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Gollege 

R.ECOR.D 

A  Journal  Devoted  to  the  Practical  Problems 
of  Elementary  and  Secondary  Education 
and  the  Professional  Training  of  Teachers 


March,  1903 


MATHEMATICS 


IN  THE 
ELEMENTARY  SCHOOL 


IAY  1917 


PUBLISHED   BY 

THE  COLUMBIA   UNIVERSITY   PRESS 

Columbia  University,  New  York 
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(COPYRIGHT,  1900,  BY  TEACHERS  COLLEGE) 


Teachers  College  Record 

Edited  by  DEAN  JAMES  E.  RUSSELL 

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TEACHERS  COLLEGE  RECORD  is  a  serial  publication  issued  by 
Teachers  College,  Columbia  University,  for  the  purpose  of  pre- 
senting to  students  of  Education,  and  to  the  public  generally,  a 
comprehensive  view  of  the  history  and  principles  of  education, 
of  educational  administration,  and  of  the  theory  and  practice  of 
teaching  as  advocated  and  followed  by  Teachers  College  and  its 
schools  of  observation  and  practice. 

ANNOUNCEMENT  OF  VOLUME  IV  — 1903 

The  May  issue  of  the  TEACHERS  COLLEGE  RECORD  will  deal 
with  "  Methods  of  Teaching  French  and  German,"  according  to 
the  new  or  reform  method,  by  Dr.  Leopold  Bahlsen,  of  Berlin, 
Lecturer  on  Methods  of  Teaching  Modern  Languages,  Teachers 
College. 

Other  issues  will  discuss  the  principles  which  should  govern 
the  curriculum  of  primary  and  secondary  schools,  illustrated  from 
outlines  of  courses  as  conducted  in  the  Teachers  College  Schools, 
and  followed  by  special  issues  dealing  with  specific  work  in  the 
first,  second,  third,  and  fourth  grades,  and  in  the  high  school. 

Numbers  concerning  Kindergarten  work  and  Music  in 
Schools  may  also  be  expected. 

Address  all  communications  to 

The  Editor  of  Teachers  College  Record 

Columbia  University,  New  York  City 


Q  A 


TEACHERS  COLLEGE  RECORD 


VOL.  IV  MARCH,    1903  No.  2 


1917 
MATHEMATICS  IN  THE  ELEMENTARY  SCHOOL 

By  DAVID  EUGENE  ,SMITH,  PH.D.,  Professor  of  Mathematics,  and 
FRANK  M.  McMURRY,  PH.D.,  Professor  of  Elementary  Education,  in 
Teachers  College,  Columbia  University 

The  following  outline  of  theory  and  of  subject-matter  is  pro- 
posed rather  as  a  basis  for  discussion  with  students  in  profes- 
sional courses  than  as  a  fixed  body  of  thought  for  use  in  the 
elementary  school.  Yet  the  course  of  work  here  outlined  is  fol- 
lowed to  -a  very  large  extent  in  Teachers  College  in  its  Horace 
Mann  School  of  observation  and  its  Speyer  School  of  practice, 
although  the  arrangement  of  topics  is  necessarily  different  in  the 
two  schools.  The  particular  order  here  suggested  is  expected  to 
apply  more  fully  to  the  Horace  Mann  School  than  to  the  Speyer 
School. 


I.     CONTROLLING   IDEAS   THROUGHOUT    THE 
CURRICULUM 

The  numerous  changes  in  the  curriculum  of  mathematics  in 
the  grades  during  recent  years  have  been  principally  due  to  two 
causes,  namely,  a  growing  sympathy  with  children,  and  a  more 
enlightened  interpretation  of  the  needs  of  society.  The  former 
has  led,  in  many  schools,  to  the  omission  or  neglect  of  number 
work  as  a  regular  study  during  the  first  year  or  two  of  the  course, 
on  the  plea  that  it  is  somewhat  foreign  to  child  nature ;  it  has 
demanded  less  abstract  problems,  in  order  that  truer  interest 
91]  i 


2  Teachers  College  Record  [92 

might  be  aroused;  and  it  has  opposed  especially  long  and  diffi- 
cult solutions,  as  in  equation  of  payments  and  compound  interest, 
on  the  ground  that  they  were  an  unnecessary  physical  tax,  besides 
giving  a  wrong  idea  of  actual  business  life.  The  second  cause 
of  change,  namely,  the  better  interpretation  of  the  needs  of 
society,  has  likewise  demanded  less  theoretical  problems,  and 
has  urged  the  entire  omission  of  topics  that  are  not  called  for 
in  the  quantitative  relations  of  daily  life;  as,  for  example,  the 
table  of  troy  weight,  partnership,  and  cube  root. 

The  question  arises:  Should  these  two  ideas  constitute  the 
principal  standard  in  the  selection  of  a  curriculum  in  mathe- 
matics? And,  if  so,  dare  they  have  fuller  control  than  they  have 
thus  far  obtained?  Or,  if  neither  of  these  should  control,  what 
standard  should  take  their  place? 

A  full  discussion  of  these  questions  is  out  of  place  here, 
because  it  is  a  general  educational  problem  that  is  involved, 
applying  alike  to  all  branches  of  knowledge,  and  in  the  main 
it  should  be  assumed  as  settled,  when  an  individual  study  is 
under  consideration.  Nevertheless,  a  brief  mention  of  the  former 
standard  of  selection  of  materials  in  mathematics  for  children, 
and  a  comparison  of  the  same  with  the  generally  accepted 
standard  in  certain  other  studies,  may  suggest  our  answer  to  the 
proposed  questions,  and  an  outline  of  our  argument. 

Twenty  years  ago,  acquaintance  with  arithmetical  processes, 
and  mental  discipline,  constituted  the  chief  purposes  of  instruc- 
tion in  arithmetic.  As  to  subject-matter,  whatever  problems 
promised  best  to  acquaint  pupils  with  these  useful  processes, 
and  to  furnish  this  mental  training,  were  deemed  acceptable. 
This  is  probably  still  the  prevalent  view. 

But  these  aims,  or  standards  of  worth,  are  far  from  accept- 
able in  certain  other  studies.  Turn,  for  example,  to  literature,  his- 
tory, nature  study,  and  art,  as  carried  on  in  the  grades.  Who  would 
rank  business  utility  and  mental  discipline  foremost  among  their 
purposes,  and  be  guided  primarily  by  these  objects  in  the  selec- 
tion of  topics  in  those  subjects?  Indeed,  the  whole  point  of  view 
has  radically  changed  as  to  these  branches.  The  teacher's  first 
)  aim  is  the  excitement  of  a  deep  interest,  possibly  love,  for  these 
fields  of  human  experience.  That  is  the  important,  immediate, 
goal  to  be  reached.  When  it  comes  to  the  subject-matter,  those 
topics  must  be  chosen  which  are  capable  of  arousing  interest; 


93]  Controlling  Ideas  Throughout  the  Curriculum  3 

this  is  one  of  the  controlling  ideas,  and  the  teacher  must  become 
a  close  student  of  children  in  order  to  avoid  serious  error  at  this 
point. 

But  not  everything  that  is  thoroughly  interesting  can  be 
accepted,  and  there  is,  therefore,  a  second  controlling  idea  for 
selection.  The  subject-matter  in  each  of  those  studies  must 
reveal  some  side  of  life,  and  do  it  in  such  a  way  that  the  pupil 
feels  forcibly  the  relation  between  it  and  practical  living.  The 
truths  of  literature  must  offer  a  guide  for  daily  action;  those  of 
history  must  throw  light  upon  and  awaken  sympathy  for  present 
social  problems ;  nature  study  must  lead  into  agriculture ;  and 
art  must  end  in  the  real  enjoyment  of  such  pictures  as  should 
grace  the  home,  as  well  as  in  the  desire  and  ability  to  make  home 
more  attractive.  Whatever  matter  contributes  only  slightly  to 
this  purpose  of  richer,  more  effective  present  living,  must  go; 
hence,  non-classics  in  literature,  numerous  names  and  dates  in 
history,  mere  sense-training  object-lessons  "  of  any  sort "  in 
nature  study,  and  the  bare  mechanical  drawing  of  geometric 
forms  in  art,  are  discarded.  In  other  words,  an  actual  weeding-; 
out  process  is  going  on  in  these  parts  of  our  curriculum,  with^ 
the  object  of  searching  out  and  retaining  only  those  materials 
that  (a)  correspond  with  child  nature,  and  (b)  identify  the  child 
with  actual  life. 

If  these  large  purposes  are  accomplished,  minor  objects  are 
also  attained,  including  such  as  were  above  mentioned  in  con- 
nection with  arithmetic.     For  instance,  when  genuine  interest  is 
assured,  the  prime  condition  for  concentration  of  attention  is  ful-1 
filled ;  hence,  good  mental  discipline  is  bound  to  follow,  provided,  j 
of  course,  the  method  of  presenting  the  subject   is   good.     In' 
brief,  good  mental  discipline,  as  an  aim,  has  little  to  do  with  the 
choice  of  subject-matter;  it  is  thought  of  only  after  this  matter 
has  been  chosen,  and  in  that  sense  is  a  subsidiary  object.     Who, 
for  example,  would  allow  the  development  of  logical  power  to 
be  a  controlling  factor  in  selecting  literature  and  history  topics? 
And  even  if  it  were  so  allowed,  how  could  it  be  a  clear  guide? 

Again,  the  business  utility  of  a  subject  is  of  value;  but  if 
history  leads  into  the  heart  of  present  social  problems,  if  nature 
study  opens  to  view  many  broad  principles  of  agriculture,  and  if 
art  makes  the  home  more  attractive,  a  good  degree  of  business 
utility  is  attained,  and  much  more.  Utility  taken  in  a  narrow 


4  Teachers  College  Record  [94 

sense  kills  the  spirit  of  any  study,  as  we  all  know ;  but  when 
each  of  these  branches  aims  in  this  broad  way  to  meet  the  needs 
of  life,  not  only  useful  knowledge  is  acquired,  but  spirit  and 
energy  are  aroused. 

Enough  has  been  said  to  indicate  our  answer  to  the  query 
raised  at  the  beginning  of  this  article.  Business  utility  and 
mental  discipline  should  not  rank  as  the  primary  aims  in  teaching 
mathematics  to  children ;  particularly  in  the  selection  of  subject- 
matter.  This  study  should  stand  on  the  same  plane,  should  be 

controlled  bv  the  same  broad  ideas,  as  other  studies.     Accord- 
'  .         /  . 

ingly,(the  child's  interest  in  the  quantitative  side  of  life  should 

be  the  highest  immediate  aim  of  the  teacher  of  mathematics  in 
the  gradesy  just  as  his  interest  in  the  spiritual  side  is  the  highest 

"  immediate  aim  of  the  teacher  of  literature;  and  the  nature  of  the 
child,  together  with  the  needs  of  society,  should  constitute  the 
main  standard  in  selecting  subject-matter.  This  signifies  that 
these  two  ideas  should  be  given  a  fuller  control  in  mathematics 
than  has  thus  far  been  allowed. 

What  is  there,  however,  in  mathematics  in  the  grades  that 
is  capable  of  appealing  to  interest?  Has  it  a  body  of  thought 
comparable  in  attractiveness  to  children  to  that  in  literature  or 
history  or  nature  study?  That  is  the  most  fundamental  question 
in  this  field  at  the  present  time.  If  there  is  no  such  body  of 
thought,  then  of  course  we  must  drop  back  upon  useful  knowl- 
edge and  mental  discipline  as  our  chief  aims.  To  be  sure,  these 
are  purposes  that  have  been  "  conceived  by  the  adult  and  forced 
upon  the  child,"  but  what  else  would  be  left  ?  On  the  other  hand, 
if  there  is  a  rich  body  of  thought  here  for  the  pupil,  then  it  is 
time  that  we  were  finding  it  out. 

Manual  training  has  faced  this  same  question  and  is  in 
advance  of  arithmetic  in  its  solution.  It  was  not  long  ago  that 
this  study,  like  arithmetic,  aimed  mainly  at  utility,  by  leading 

"  children  to  make  useful  objects,  and  at  mental  discipline,  by 
exercising  the  mind  through  the  hand.  But  now  there  is  a 
marked  tendency  to  regard  handwork  as  a  means  of  opening  up 
the  industrial  side  of  life.  Through  it,  one  is  not  only  to  develop 
some  skill  in  carpentry,  blacksmithing,  masonry,  basketry,  and 
bent-iron  work;  but  he  is  to  be  introduced  into  the  constructive 
experiences  of  mankind,  making  possibly  many  excursions  to 
factories,  in  order  the  more  fully  to  comprehend  and  sympathize 


95]  Controlling  Ideas  Throughout  the  Curriculum  5 

with  this  phase  of  human  activity.  And  why  is  this  not  reason- 
able? Just  as  literature,  relying  upon  words,  tries  to  present 
man's  highest  aspirations,  why  should  not  some  other  study, 
relying  largely  upon  observation  and  participation  in  manufac- 
turing, try  to  present  man's  constructive  experience?  Little 
seems  to  be  lost,  and  much  gain  is  promised,  by  this  change;  for 
the  thought-content  of  manual  training  is  thereby  increased  far 
beyond  the  principles  involved  in  the  use  of  tools.  The  fact  that 
we  are  distinguished  as  an  industrial  nation  emphasizes  the  need 
of  this  advance,  on  the  social  side,  while  it  promises  an  especially 
rich  addition  to  the  subject-matter  of  manual  training.  When 
this  aim  is  more  fully  realized  in  practice,  as  it  can  be,  manual 
training  will  stand  on  the  same  educational  plane  as  other  studies, 
and  the  instructor  in  "that  branch  can  then  justly  rely  upon  im- 
planting a  permanent  interest  in  it,  as  do  instructors  of  literature 
and  nature  study  in  their  respective  fields. 

Again  the  question  is  before  us :  What  is  there  in  arith- 
metic that  is  capable  of  arousing  the  direct,  immediate,  interest 
of  children?     Ordinarily  the  part  of  any  study  that  is  depended 
upon  to  arouse  interest,  at  least  initial  interest,  is  the  part  that' 
is  concrete,  as  the  individual  facts  in  history  and  nature  study. 
The  mere  operations  in  arithmetic  are  abstract,  as  we  know,  and 
the  principal  other  part  of  the  study  is  the  applied  problems. 
Hence,  these  applied  problems  must  be  the  concrete  part  of  the, 
subject,  and  if  its  content  is  to  be  greatly  enriched,  the  change 
must  be  expected  first  of  all  in  this  particular. 

But  some  persons,  no  doubt,  will  here  raise  the  questions: 
Are  the  problems  of  arithmetic  really  concrete?  Has  arithmetic, 
in  fact,  any  concrete  part  at  all,  as  other  studies  have? 

It  becomes  necessary  to  define  what  we  mean  by  concrete. 
Some  years  ago  it  was  found  that  a  very  large  percentage  of 
examples  in  our  arithmetics  were  plainly  abstract,  consisting 
of  unnamed  units,  such  as  5^2X3%-  To  make  these  concrete 
they  were  made  denominate;  as,  for  example,  If  one  barrel  of 
apples  cost  $3^,  what  will  5^2  barrels  cost?  But  still  there  is 
the  common  complaint  that  these  are  practically  as  abstract  as 
they  were  before.  When,  therefore,  is  an  example  concrete? 

If  one  goes  to  a  dictionary  to  learn  the  meaning  of  a  list 
of  new  words,  he  is  engaged  in  an  abstract  kind  of  work, 
because  the  words  are  then  dissociated  from  their  original,  con- 


6  Teachers  College  Record  [96 

crete,  connections.  If,  on  the  other  hand,  one  learns  their  mean- 
ing through  the  context,  as  most  children  do,  the  work  is 
concrete,  because  the  words  are  then  approached  and  viewed  in 
their  individual  relations.  So,  in  general,  a  statement  is  truly 
concrete  only  when  it  is  seen  in  its  original  associations;  the 


moment  it  is  "  isolated  from  its  origin  and  its  goal,  it  becomes 
abstract." 

Applied  to  mathematics,  this  means  that  problems  are 
abstract  when  they  are  chosen  at  random,  without  the  purpose 
of  clarifying  the  individual  situations  in  which  they  arose,  but 
merely  with  the  intent  of  giving  exercises  in  processes.  They 
^-are  concrete  only  when  they  deal  with  real  things  and  with  actual, 
significant,  situations.  In  the  former  case,  the  answers  reached 
have  no  worth  in  themselves,  and  the  work  is  unreal  because 
the  processes,  which  were  originally  a  mere  means,  become  the 
end.  Motive  for  hard  work  on  the  part  of  pupils  is  no  longer 
provided  in  the  subject  matter,  and  for  that  reason  competition 
of  child  with  child,  an  artificial  incentive,  must  be  largely  relied 
upon.  In  the  latter  case  the  answers  are  the  end,  because  they 
'  are  valuable  thoughts,  like  those  in  other  studies. 

For  example,  let  us  take  for  our  topic  a  comparatively 
insignificant  industry,  namely,  the  manufacture  and  consumption 
of  printers'  ink  in  the  United  States : 

1.  At  the  present  time  about  30  million  pounds  of  this  ink  are  manu- 
factured in  the  United  States  per  year.    How  many  tons  is  that?    {Answer, 
15,000.) 

A  single  factory  in  New  York  makes  il/2  million  pounds.  What  part 
of  the  whole  does  it  make?  (V».)  How  many  tons?  (750.) 

2.  The  worth  of  this  one  plant  is  about  $125,000.     At  that  rate,  how 
much  money  is  invested  in  this  industry  in  the  United  States?  ($2,500,000.) 

3.  This  one  factory  employs  20  workmen,  and  35  men  in  all,  including 
salesmen  and  others.    What,  then,  are  the  corresponding  numbers  employed 
in  the  entire  industry?     (700.) 

4.  The  workmen  in  this  factory,  on  the  average,  receive  about  $2  per 
day.    At  that  rate  what  amount  is  paid  per  day  to  such  workmen  in  the 
United  States?    Per  year,  of  300  days?    The  35  men  average  about  $5  per 
day.    What  would  be  the  approximate  pay  roll  per  week,  and  per  year,  for 
all  such  factories  together? 

5.  This  one  factory,  in  making  the  ink,  consumes  per  month  250  bbls. 
of  rosin  brought  mainly  from  the  pine  trees  of  Georgia  and  South  Caro- 
lina.   It  costs  $6  per  bbl.    What  facts  can  you  figure  out  from  these  state- 
ments?    (3,000  bbls.  are  used  in  the  one  factory,  and  they  cost  $18,000. 
The  cost  for  all  the  factories  is  $360,000.) 


97]  Controlling  Ideas  Throughout  the  Curriculum  7 

6.  This  one  factory  consumes  per  month  10,000  gal.  of  linseed  oil, 
made  from  flax  seed,  and  worth  75  cents  per  gal.     What  facts  can  you 
learn  from  this  statement?     (It  costs  the  one  factory  $7,500  per  month,  or 
$90,000  per  year,  for  such  oil.    It  costs  all  the  factories  $1,800,000  per  year, 
the  total  amount  being  2,400,000  gal.     The  acreage  of  flax  might  also  be 
considered.) 

7.  Four  hundred  tons  of  anthracite  coal  are  burnt  per  year  in  tnis 
factory.     What  is  its  price,  at  the  present  rate? 

8.  This  factory  consumes  600  bbls.  of  water  per  day,  300  days  in  the 
year,  mainly  to  keep  cylinders  cool.     The  city  charges  $i  per  thousand 
cu.  ft.  of  water.    The  barrels  contain  50  gal.  each,  and  I  gal.  of  water  weighs 
8  Ibs.    A  cubic  foot  of  water  weighs  62%  Ibs.    The  factory  has  recently 
dug  a  well  of  its  own.     Estimate  how  much  it  saves  by  that  means,  sup- 
posing its  pumping  to  cost  nothing. 

9.  One  and  one-fourth  Ibs.  of  black  ink  will  print   1000  eight-page 
newspapers  of  ordinary  size.  How  much  would  an  eight-page  weekly  paper, 
having  a  circulation  of  12,000,  consume  per  year?    Find  out  the  circulation 
of  your  principal  county  paper,  and  compute  the  amount  of  ink  it  uses. 

10.  How  much  would  be  consumed  by  one  issue  of  the  New  York 
Morning  Journal,  a  sixteen-page  paper,  having  a  circulation  in   1898  of 
309,472  ?    One  issue  of  the  New  York  Sunday  World,  a  fifty-six-page  paper 
with  a  circulation  in  1898  of  500,000?     (2  tons  and  375  Ibs.  for  the  World.) 

11.  The  retail  price  of  black  ink  varies  from  4  to  10  cts.  per  Ib.    At 
6  cents  per  Ib.,  how  much  would  one  of  the  above  named  papers  pay  for 
ink,  per  year?     (World,  $262.50  per  Sunday.) 

12.  The  black  part  of  black  ink  is  like  lampblack,  and  is  made  from 
gas.    One  pound  of  such  black  is  worth  8  cents;   and  it  takes  2000  cu.  ft. 
of  gas  to  make  it.     Gas  in  New  York  City  costs  $i  per  thousand  cu.  ft. 
Is  this  lampblack  probably  manufactured  in  the  city?     If  not,  can  you 
suggest  where  it  might  be  made? 

These  examples,  instead  of  being  hypothetical,  are  actual 
cases,  and  in  that  sense  are  concrete.  Some  of  the  answers 
might  well  be  worth  remembering,  like  facts  in  history.  Of 
course,  if  pupils  have  not  become  interested  in  New  York  City 
and  its  industries,  this  topic  might  be  relatively  unattractive  to 
them.  But  that  is  a  danger  encountered  in  all  studies,  and 
suggests  the  desirable  relationship  in  the  grades  between  mathe- 
matics and  other  branches.  Teachers  in  composition  and  in 
manual  training  try  to  select  those  topics  to  work  upon,  that 
have  already  become  of  interest  to  a  class.  These  are  recognized 
as  branches  of  knowledge  that  are  quite  dependent  upon  others 
for  suggestions  as  to  subject  matter;  and  the  recognition  of  this 
dependence  by  no  means  detracts  from  their  dignity.  Indeed, 
dignity  is  not  concerned  here;  this  arrangement  merely  insures 


8  Teachers  College  Record  [98 

a  greater  degree  of  interest.  Since  elementary  arithmetic  is  only 
the  quantitative  side  of  human  experience,  it  is  peculiarly  de- 
pendent upon  other  fields  for  the  content  of  its  problems,  although 
by  no  means  dependent  solely  upon  other  school  studies.  The 
more  the  problems  deal  with  interesting  topics,  the  better;  hence, 
correlation  of  the  right  kind  is  a  matter  of  supreme  importance 
to  mathematics  in  the  grades. 

Further  than  that,  the  correlation  of  mathematics  is  of  very 
great  importance  to  other  branches,  and  even  necessary  to  the 
effectiveness  of  the  curriculum  as  a  whole.  For  instance,  geog- 
raphy gives  the  causes  for  the  location  of  factories  in  a  certain 
place;  nature  study  shows  how  the  earthworm  grinds  up  soil; 
and  the  United  States  history  describes  the  campaigns  of  the 
armies  in  Virginia  during  the  Civil  War,  when  it  was  necessary 
to  change  the  base  of  supplies.  Now  each  of  these  topics  has  a 
quantitative  side,  and  the  picture  is  very  incomplete  in  each  case 
until  that  side  has  been  presented.  In  fact  the  main  part  of  the 
definition  of  factory  must  be  gotten  through  arithmetic,  as  shown 
by  the  examples  on  ink  manufacture,  just  given.  The  agricul- 
tural value  of  earthworms  begins  to  be  appreciated  only  when 
we  measure  the  quantity  of  their  castings  on  a  given  area;  and 
the  difficulties  attending  an  army's  change  of  its  base  of  supplies 
are  not  clear  until  one  knows  how  much  bread,  meat,  flour,  sugar 
and  water  100,000  men  consume  in  a  day ;  and  how  many  horses, 
wagons  and  men  it  takes  to  haul  that  amount  of  supplies  a  given 
distance  in  a  given  time.  The  single  fact  that  it  requires  not 
less  than  40  bbls.  of  salt  per  day  for  such  an  army  throws  much 
light  on  the  matter.  Figures  give  descriptions,  just  as  words  do, 
and  our  appreciation  of  the  quantitative  side  of  life  is  lamentably 
•  poor  because  we  have  made  so  little  use  of  figures  in  school  for 
this  purpose.  Indeed,  many  of  our  most  pressing  problems  are 
• — largely  quantitative,  as  for  example,  the  maintenance  of  the  Erie 
Canal  in  the  State  of  New  York.  Why  should  we  not  know 
the  relative  cost  and  speed  of  transportation  by  rail  and  canal, 
together  with  cost  of  maintenance,  as  well  as  the  parts  of  speech 
and  the  causes  of  our  wars?  And  'must  we  not  possess  such 
quantitative  knowledge  before  we  can  be  intelligent  as  to  public 
policy  in  such  matters? 

This  quantitative  knowledge,  too,  calls  for  more  than  arith- 
»  metic  alone,  even  in  childhood.  It  requires  mensuration,  some 


99]  Controlling  Ideas  Throughout  the  Curriculum  9 

geometry,  and  even  a  little  algebra.  Why  should  we  try  to 
maintain  the  barriers  between  these  fields  when  they  are  so 
intimately  related  in  daily  life,  and  so  simple  in  their  elements? 
Why  should  the  mediaeval  "  Rule  of  False  Position "  still  find 
place  in  our  arithmetics  under  the  form :  "  Let  100  per  cent 
represent  the  cost "  ?  It  is  in  every  way  simpler  to  adopt  the 
symbolism  that  for  nearly  three  centuries  has  been  conventional 
in  mathematics,  and  say :  "  Let  x  represent  the  cost."  In  either 
case  we  employ  a  symbolism,  but  the  second  one  is  the  simpler, 
the  more  easily  handled  in  the  solution,  and  hence  the  more 
widely  accepted.  Indeed,  if  we  had  adopted  the  symbolism  of 
the  simple  equation  long  ago,  in  our  own  elementary  arithmetic, 
topics  like  proportion  and  the  applications  of  percentage  would 
have  been  much  more  clearly  understood,  and  the  subjects  of 
cube  root  and  the  progressions  might  have  developed  enough 
of  value  to  have  made  it  worth  while  to  retain  them.  The  fact 
is,  there  is  no  well  defined  line  between  our  common  arithmetic 
and  algebra,  and  the  examiner  in  the  former  who  appends  to  a 
question  the  statement,  "  Do  not  solve  by  algebra,"  merely  reveals 
his  own  ignorance.  He  might  insist  that  2 :  5=7 :  ( ?)  is  a 

oc      c 
problem  in  arithmetic,  and  that  -=—  is  one  in  algebra,  but  a 

little  thought  shows  that  they  are  identical.     The  fact  is  that, 
strictly  speaking,  algebra  is  a  science  treating  of  a  certain  class  j 
of  functions,  and  that  arithmetic  is  a  science  treating  of  numerical  1 
values.     But  arithmetic  and  algebra,  as  we  commonly  speak  of 
them,  are  sciences  each  of  which  considerably  overlaps  the  other,] 
and  rightly  so. 

The  same  overlapping  is  apparent  for  arithmetic  and  geome- 
try. Geometry  is  not  merely  a  modern  form  of  Euclid,  although, 
in  England  and  America  especially,  we  are  wont  to  speak  of  it 
as  such.  The  necessary  mensuration  of  common  life  is  a  part 
of  it,  and  as  such  is  deserving  of  a  much  more  scientific  treat- 
ment than  it  has  already  had.  This  topic,  however,  is  reserved 
for  a  subsequent  article. 

From  these  statements  it  follows  that  the  subjects  to  be 
studied  in  mathematics  should  be  selected  with  the  same  care 
as  to  age  and  present  interests  of  the  children  concerned,  as 
are  classic  poems.  Answers  would  then  be  worth  something,  and 
children  would  not  need  to  be  blamed  for  "  working  for  them  " 


io  Teachers  College  Record  [100 

when  they  have  nothing  else  for  which  to  work.  Then  mathe- 
matics in  the  grades  would  be  recognized  as  revealing  one  very 
interesting  and  large  side  of  life,  as  does  any  other  study  worthy 
of  pursuit. 

And  is  this  not  natural?  Children  have  a  constant  call  for 
number  in  their  play.  In  fact,  one  reason  advanced  for  having 
no  regular  study  called  number  in  the  first  two  years  of  school  is 
that  they  learn  nearly  as  much  of  this  outside  of  school  as  in  it. 
But  it  is  not  for  the  processes  that  they  care ;  they  wish  quantitative 
facts,  and  the  answer,  if  the  problem  is  real,  is  after  all  the  true 
goal.  It  is  so  with  adults  also;  they  are  wanting  the  answer 
when  they  figure  at  all;  it  is  this  that  furnishes  the  motive  for 
accuracy.  Why,  then,  should  children  in  school  be  confined  to 
/  processes,  when  the  purpose  of  children  outside  of  school,  and 
of  adults,  is  quite  apart  from  processes? 

The  whole  history  of  printed  arithmetics  shows  how  periodi- 
cally has  arisen  the  question  as  to  the  nature  of  the  work  to  be 
demanded  of  pupils.  While  there  has  been  a  constant  evolution 
towards  a  better  treatment  of  the  subject,  a  better  understanding 
of  the  processes,  a  clearer  presentation  of  the  operations,  and  a 
recognition  of  the  more  valuable  symbols  of  mathematics,  there 
has  not  been  a  corresponding  evolution  in  the  nature  of  the 
problems.  This  has  resulted  from  the  ever  present  struggle 
between  culture  and  utility;  between"  mental  discipline  and  the 
problem  of  daily  bread;  between  aristocracy  and  trade.  This 
struggle  was  seen  in  the  earliest  printed  books,  when  the  first 
/  arithmetics  printed  in  Italy  (Treviso,  1478)  and  in  Germany 
(Bamberg,  1482)  took  the  utility  side  of  the  case;  while  the 
great  Italian  treatise  of  Paciuoli  (1494),  and  the  first  English 
arithmetic  (Bishop  Tonstall's,  1522),  took  the  culture  side.  But 
this  is  a  general  historic  truth:  No  arithmetic  has  flourished 
among  a  democratic  people,  and  but  two  or  three  have  had  any 
great  hold  in  schools  that  stood  for  pure  culture,  that  did  not 
minimize  processes  and  magnify  the  study  of  those  problems 
that  revealed  the  actual  quantitative  side  of  the  life  of  their  day. 
Thus  it  was  that  Kobel  and  Riese  in  the  i6th  century  in  Germany 
had  an  enormous  influence;  that  Trenchant  and  Savonne  at  the 
close  of  the  same  century  in  France  dominated  the  teaching  in 
that  country;  that  Recorde  did  the  same  for  England  then,  as 
Cocker  did  later;  and  that,  with  the  great  awakening  of  com- 


IQI]          Controlling  Ideas  Throughout  the  Curriculum  n 

merce  in  Holland  in  the  i/th  century,  a  flood  of  books  of  this 
class  appeared,  revealing  in  a  most  interesting  way  the  actual  life 
of  the  people  of  that  period. 

The  problems  of  present  arithmetic  might  well  be  of  two 
kinds :  f ir^t,  those  dealing  with  the  quantitative  side  of  matters1/ 
of  local  interest;  as  the  cost  of  blasting  rock  for  a  cellar  in  New 
York  City,  the  quantity  and  cost  of  a  mile  of  asphalt  pavement 
in  Buffalo,  the  quantity  of  water  necessary  for  irrigating  for  a 
season  an  acre  of  land  in  Colorado,  and  the  cost  and  quantity 
of  materials  necessary  for  fattening  a  herd  of  25  cattle  on  an 
Ohio  farm ;  second,  those  dealing  with  the  quantitative  side  of  . 
matters  of  general  interest ;  as  the  Pennsylvania  Railroad  system, 
an  ocean  steamship,  the  comparative  cost  of  transportation  of  ore 
from  Lake  Superior  to  Pittsburg  by  rail  and  by  water,  the 
amount  of  freight  carried  on  the  Mississippi  compared  with  that 
carried  by  the  Illinois  Central  Railroad  which  parallels  it,  a  great 
department  store;  the  labor  and  money  saved  by  the  cotton  gin 
and  by  other  inventions  concerned  in  cotton  production;  the 
sugar-cane  and  beet-sugar  industries  compared;  and  dairying 
and  ranching.  Since  nearly  all  of  the  arithmetical  processes 
are  mastered  by  the  end  of  the  fifth  year  at  school,  the  last 
three  years  of  mathematics  in  the  grades  might  be  spent  al- 
most entirely  in  a  study  of  important  industries  and  other  •> 
matters  on  the  quantitative  side:  as  mining,  banking,  invest- 
ments, manufacture  of  clothing,  government  revenues,  commis- 
sion business,  gardening  and  farming,  the  cost  of  paving,  of 
water  and  of  gas  in  different  cities,  the  comparative  cost  of  gas 
and  electric  light,  a  comparison  of  the  steel  industry  in  this 
country  and  in  England,  or  a  comparison  of  the  growth  of  cer- 
tain cities  at  home  and  abroad. 

Such  study  would  allow  mathematics  to  culminate  in  two 
classes  of  generalizations :  ( I )  the  rules  of  arithmetic  to  which 
we  are  already  accustomed,  and  (2)  another  large  class  of  truths 
dealing  with  the  material  side  of  human  affairs. 

Furthermore;  consider  how  such  arithmetic  would  meet  the 
"  needs  of  society,"  since  the  subject-matter  leads  so  directly 
into  practical  life.  The  relation,  too,  may  be  one  of  sympathy 
as  well  as  of  knowledge.  A  New  York  street  car  conductor  very 
often  rides  80  miles  per  day.  When  we  recall  this  fact,  together 
with  the  number  of  fares  collected,  the  number  of  stops  made, 


12  Teachers  College  Record  [102 

the  number  of  persons  helped  on  and  off,  and  the  number  of 
hours  consumed  by  all  this,  his  life  touches  ours  as  it  never  did 
before;  and  this  closer  union  means  deeper  sympathy.  When, 
through  actual  data,  we  learn  that  the  owner  of  city  houses  to 
rent  must  count  upon  their  being  idle  from  one-tenth  to  one- 
quarter  of  the  time,  we  understand  better  the  reason  for  his  high 
prices.  And  when  again,  from  actual  facts,  we  know  that  even 
a  careful  grocer  in  a  city  expects  to  have  15  per  cent  of  his 
fresh  fruit  spoil  in  hot  weather,  and  to  lose  one-tenth  of  all  the 
money  that  he  credits,  we  are  at  least  able  to  understand,  if  not 
always  to  condone,  some  of  his  actions. 

The  fact  that  the  answers  to  many  of  these  actual,  living, 
problems  can  be  only  approximately  correct,  shows  the  necessity 
for  a  class  of  examples  that  do  not  "  come  out  even,"  a  class  that 
really  meet  the  "  needs  of  society."  A  large  percentage  of  busi- 
ness is  of  necessity  merely  a  matter  of  estimates ;  while  even  in 
/the  most  careful  scientific  experiments,  the  data  can  never  be 
exact,  and  the  results  of  computations  cannot  be  closer  to  the 
truth  than  are  these  data. 

It  is  by  such  a  plan  as  the  preceding  that  mathematics  in 
.  the  grades  might  accept  the  "  interest  of  the  child  "  and  the  "  re- 
quirements of  society  "  as  its  controlling  ideas  in  the  selection 
I  \  'of  subject-matter.  And,  under  such  control,  the  utility  of  the 
knowledge  and  the  value  of  the  mental  discipline,  using  these 
terms  in  the  common  sense,  would  be  enhanced  rather  than 
diminished ;  while  much  higher  purposes  in  the  study  would  also 
be  accomplished. 

But  the  above  plan  is  far  from  practicable  when  present 
conditions  are  considered.  In  mathematics  there  must  be  more 
dependence  upon  text-books  than  in  most  other  studies,  and  our 
text-books  have  been  prepared  along  traditional  lines.  We  can 
omit  obsolete  matter,  but  satisfactorily  to  add  such  topics  as 
some  of  those  mentioned,  and  to  rearrange  the  whole  on  newer 
lines,  is  impossible  without  much  time  and  effort. 

The  difficulty  is  particularly  noticeable  in  the  sixth,  seventh, 
and  eighth  grades.  The  children  there  know  the  fundamental 
operations  with  both  integers  and  fractions,  and  should  be  ready 
'  to  devote  all  their  time  to  applications  needed  in  modern  life, 
and  the  facts  that  arise  from  such  application.  But  the  arith- 
metics, instead,  omit  this  latter  side  and  often  offer  applications 


103]         Controlling  Ideas  Throughout  the  Curriculum  13 

of  past  generations.      For   instance,   they   elaborate   proportion, 
which  is  a  mediaeval  method  of  solving1  simple  equations,  and 
neglect  the  geometric  application  to  measuring   figures  of  the  * 
same  shape  (see  p.  62),  which  is  really  about  the  only  application 
of  importance  outside  of  physics. 

Under  the  present  conditions,  however,  it  is  impossible  to 
do  more  than  keep  clearly  in  mind  the  plan  which  we  should  like 
to  follow,  and  to  struggle  toward  it  as  best  we  can,  here  and 
there,  in  practice.  It  should  be  remembered,  also,  that  it  is  the) 
purpose  of  this  publication  to  suggest  suitable  controlling  ideas 
and  a  corresponding  outline  of  subject  matter,  rather  than  to  be 
a  text-book  in  itself. 

In  conclusion,  it  is  the  mission  of  mathematics  in  the  grades 
to  make  a  large  contribution  to  the  knowledge  of  the  children, 
and  to  appeal  to  their  interest  by  the  richness  of  its  content. 
Therefore  processes  can  be  only  a  subordinate  part  of  the  subject- 
matter.  Deep  interest,  surprise,  and  excitement  should  be  pro- 
duced by  the  valuable  thoughts  in  this  subject  as  in  others. 
There  is  less  danger  in  the  criticism  that  an  arithmetic  is  too 
much  like  an  encyclopaedia,  than  in  the  criticism  that  it  is  too 
much  like  a  dictionary. 


II.     OUTLINE  FOR  THE  FIRST  FIVE  GRADES 

Grade  I 

I.      General   Suggestions 

(a)  In  the  spirit  of  the  foregoing  article,  the  purpose  of 
mathematics  for  six-year-old  children  is  to  meet  from  their 
point  of  view,  their  daily  need  of  number  as  it  arises  in  their 
school  studies  and  in  their  relations  outside  of  school.  And  lest 
it  may  be  felt  that  we  are  emphasizing  the  abstract,  in  speaking 
of  number  rather  J:han  measurement,  of  which  latter  we  have 
recently  been  hearing  so  much,  some  explanation  should  be  given. 
To  a  child,  as  to  us,  to  measure  anything  is  to  count  the  number 
of  times  some  arbitrary  unit  is  contained  in  that  thing.  To  find 
a  ratio  of  one  thing  to  another  is  to  find  how  many  times  the 
one  contains  the  other,  or  what  part  it  is  of  the  other;  or, 
what  is  the  same  thing,  to  find  how  many  times  each  contains 
some  common  unit.  Hence  the  somewhat  persistent  argument 
that  we  should  teach  a  child  measurement  rather  than  number, 
or  ratio  as  the  basis  of  number,  is  simply  to  say  that  we  should 
teach  number  with  due  regard  to  its  applications.  To  this  we  have 
already  agreed;  but  to  confine  the  early  study  of  number  to  the 
measurement  of  lengths  alone,  and  ratio  to  the  consideration  of 
an  uninteresting  series  of  blocks,  is  entirely  foreign  to  our  belief 
as  expressed  in  the  preceding  article  as  to  what  arithmetic,  at 
any  stage,  should  be.  Having  now  explained  our  understanding 
of  the  relation  of  number  to  measurement,  we  return  to  the 
work  of  the  first  grade. 

Systematic  instruction  in  this  subject  at  this  age  is  extremely 
difficult,  owing  to  the  danger  of  its  being  too  formal.  On  that 
account,  and  also  because  of  their  peculiar  need  of  other  kinds 
of  work,  the  children  of  the  Speyer  School  should  probably 
have  no  regular  study  of  mathematics  during  the  first  school 
year.  On  the  contrary,  it  should  be  quite  incidental.  But, 

14  [104 


105]  Outline  for  the  First  Five  Grades  15 

as  they  are  using  a  large  number  of  materials  that  require  some 
kind  of  measurement,  they  will  incidentally  acquire  much  knowl- 
edge in  this  field,  provided  the  teacher  is  fairly  attentive  to  the 
quantitative  side  of  their  experience. 

The  children  of  the  Horace  Mann  School,  on  the  other  hand, 
begin  the  course  with  more  mature  minds,  owing  to  their  home 
training.  They  have  heard  more  of  business  in  a  large  way, 
they  have  come  more  closely  in  contact  with  nature,  they  have 
traveled  and  have  listened  to  lively  table  conversation,  and  they 
can  more  safely  run  the  danger  of  formal  work.  For  them  some 
regular  study  in  this  line  is  not  only  safe  but  valuable.  But 
while  the  recitation  period  should  be  regular,  the  body  of  thought 
offered  should  make  no  attempt  to  constitute  a  system,  such,  for 
example,  as  that  outlined  by  the  "  Grube  Method."  Such  a  plan 
is  the  logical  arrangement  of  the  adult  mind,  and  ignores  the  need 
of  motive  on  the  part  of  the  child.  The  two  schools,  therefore, 
cannot  cover  the  same  ground  either  during  the  first  school  year, 
or  later.  The  following  outline  suggests  the  quality  and  arrange- 
ment of  work;  each  of  the  schools  should  do  as  much  or  as 
little  as  is  fitting.  And  what  is  true  of  these  schools  is  true  of 
others.  It  is  always  dangerous  to  lay  down  a  strict  programme 
for  several  schools ;  some  latitude  is  always  necessary.  Even 
more  dangerous  is  a  series  of  books  definitely  limited  as  to  grade 
work. 

(&)  Desirable  Materials.  The  materials  needed  are:  foot 
rules,  yard  sticks,  i-lb.  weights  with  balance  scales  for  weighing, 
toy  or  real  money,  pint  and  quart  measures,  building  blocks  of 
definite  dimensions,  including  many  inch  cubes ;  Speer  blocks, 
splints,  materials  for  number  games,  and  cardboard  for  making 
furniture  to  a  scale.  The  use  of  fingers  for  counting  should  be 
discouraged,  for  the  reason  that  they  cannot  later  be  removed 
entirely  from  reach,  when  not  wanted.  The  same  objection 
applies  to  the  habit  of  making  marks.  But  splints  and  other 
objects  can  be  so  removed  and  hence  are  unobjectionable.  While 
materials  should  vary,  in  order  to  hold  the  interest,  and  should 
not  lack  esthetic  value,  they  should  not  prove  so  attractive  as 
to  draw  attention  away  from  the  number  work.  Much  of  the 
gaudily  colored  material  often  sold  is  objectionable  on  this 
ground. 

(c)  Variety  is  itself  worthy  of  being  regarded  as  a  principle 


1 6  Teachers  College  Record  [106 

of  education,  and  since  very  young  children  quickly  tire  of  a 
single  kind  of  work,  it  is  important,  where  mathematics  is  a 
regular  study,  frequently  to  vary  the  materials  and  the  nature 
of  the  problems. 

(d)  Normally  in  mathematics,  written  work  should  supple- 
ment oral  work,  by  taking  up  those  problems  that  are  too  diffi- 
cult for  oral  treatment.     Written  numbers  should  receive  very 
little  attention  in  this  grade,  and  should  be  undertaken  mainly 
for  the  sake  of  variety  and  for  exercise  in  penmanship. 

(e)  Accuracy,  within  a  fraction  of  the  large  units  used,  is 
especially  important  at  this  age;  it  not  only  impresses  the  basal 
units  of  measurement  on  the  mind,  but  is  necessary  for  a  proper 
appreciation  of  the  spirit  of  this  branch  of  knowledge.     First 
impressions  are  peculiarly  important  in  the  attainment  of  this 
latter    aim.      Accuracy    is    much    more    important    than    speed, 
although  it  should  be  remembered  that  accuracy  within  small 
fractions  of  the  units  used  finally  becomes  impossible,  and  the 
attempt  to  reach  it  in  measuring  is  injurious  physically. 

(f)  A  text-book   is   unnecessary   at  this   age,   although   it 
might  secure  additional  variety  in  the  Horace  Mann  School  and 
prove  helpful  in  assigning  very  simple  kinds  of  seat  work.     On 
the  whole,  however,  it  is  likely  to  increase  the  danger  that  the 
work  will  become  too  formal. 

(g)  A  reasonable  effort  should  be  made  to  develop  the  num- 
ber sense  of  each  pupil;  but  such  effort  should  stop  short  of 
persecution  both  in  this  and  in  higher  grades,  even  though  some 
children  are  thought  to  show  "  no  ability  to  learn  mathematics." 
Excellence  in  every  study  is  not  a  sine  qua  non  in  the  Elemen- 
tary School. 

(h)  One  of  the  principles  of  education  especially  applicable 
at  this  stage,  though  especially  neglected  until  recently,  is  that 
of  motor  activity.  Its  application  is  called  for  both  by  the 
physical  activity  of  the  child,  and  by  the  nature  of  the  subject, 
which  requires  actual  measurement. 

2.     The  Mathematical  Work. 

(a)  The  number-space  is  limited,  extending  from  I  to  100. 
The  chief  interest  which  children  have  in  arithmetic,  on  entering 
school,  is  in  counting,  and  this  within  the  number-space  above 
indicated. 


107]  Outline  for  the  First  Five  Grades  17 

(b)  Counting:  —  Since  much  interest  lies  in  counting,  as 
seen  in  children's  "  counting  out,"  "  keeping  score  "  in  games, 
counting  by  5's  and  by  ID'S,  in  being  "  It "  in  other  games,  and 
in  referring  to  pages  of  their  books,  special  attention  is  given 
to  the  number  series  in  the  space  I  to  100.     The  children  are 
encouraged  to  count  rapidly  by  units,  and  by  the  groups  2,  5, 
10.     The  material  objects  of  counting  should  vary  so  as  to  show 
a  wide  range  of  application.     The  emphasis  laid  upon  counting 
appeals  not  only  to  the  interest  of  the  child,  but  it  was  historically 
the  first  stage  in  the  development  of  the  world's  mathematics; 
and  it  is  in  harmony  with  the  latest  stage,  which  looks  upon 
mathematics   as   the   science  of  order,   rather  than   the   science 
of  quantity. 

In  connection  with  writing,  the  common  notation  to  20 
should  be  taught,  with  incidental  use  of  written  numbers  to  100. 
On  account  of  the  clock-face,  the  Roman  numerals  to  XII  are 
taught  in  the  second  half-year. 

The  number  idea  precedes  the  symbol  in  the  space  I  to  10. 
When  the  value  of  symbols  is  somewhat  appreciated,  the  symbol 
properly  becomes  the  more  important  of  the  two. 

(c)  Measuring:  —  All  number  is,  of  course,  the  result  of 
measure;  and  as  already   suggested,  there   is   no  dividing  line 
between  counting  and  measuring.     When  the  child  counts  the 
number  of  inches  in  a  foot,  he  has  measured  the  foot.     Hence 
measuring  appeals  to  his  interests  and  needs  at  this  time,  and 
should   form  a  considerable  part  of  the  work.     The  measures 
chiefly   used   in  this   grade,   and   with   which  the   child   should 
become  familiar  by  frequent  actual  use,  are  the  following: 

Length,  —  the  inch,  foot,  yard. 

Capacity,  —  the  pint,  quart,  gallon ;  the  quart,  peck,  bushel. 

Weight,  —  the  ounce,  pound. 

Time,  —  the  day,  week. 

The  rod  and  mile  are  omitted,  in  this  grade,  because  they 
are  not  within  the  child's  range  of  interest.  Similarly  the  gill, 
ton,  month,  year,  second,  minute  (as  1/60  of  an  hour).  But  of 
course  no  teacher  should  feel  limited  to  the  terms  given;  ex- 
pressions like  "  5  minutes,"  "  2  miles,"  and  "  6  square  inches," 
may  well  be  used  whenever  the  need  arises.  Geographical  con- 
siderations determine  many  questions  of  this  kind ;  a  child  in 
the  country  being  much  more  apt  to  know  the  width  of  a  street 


i8  Teachers  College  Record  [108 

in  rods  than  a  city  child,  and  similarly  such  measures  as  the  peck 
and  bushel. 

The  metric  system  is  not  introduced  in  the  primary  grades, 
because  it  is  not  the  one  that  children  of  this  generation  will 
chiefly  use.  It  is  needed  for  general  information  later  in  the 
course,  but  it  would  be  a  mistake  to  have  it  interfere  with  the 
common  system  here. 

(d)  Fractions:  —  As  a  result  of  attempts  to  measure,  the 
fraction  appears.     It  is  seen  in  paper-folding,  in  the  separating 
of  groups  of  objects  —  as  half  the  class,  a  quarter  of  the  blocks, 
and  in  such  other  comparisons  as  the  length  of  one  stick  compared 
with  that  of  another.    All  this  involves  the  idea  of  ratio,  a  fun- 
damental notion  in  dealing  with  number,  but  one  which  it  is 
neither  necessary  nor  advisable  to  make  very  prominent,  per  se, 
with  children. 

The  fractions  with  which  children  should  become  familiar 
in  this  grade  are  ^2,  Yz,  %,  Y±-  This  does  not,  however,  exclude 
the  incidental  use  of  such  other  fractions  as  may  naturally  enter 
into  the  work  of  the  class;  for  if  the  child  knows  thirds,  he 
knows  Y$  and  Vs  as  weN  as  Y$' 

(e)  Operations :  —  The  only  operation  to  which  much  atten- 
tion need  be  given  in  this  grade  is  addition,  and  this  only  as 
it  is  necessary  in  such  problems  as  "  5  inches  and  4  inches  are 
how  many  inches?"  or  "2  ft.,  3  ft.,  and  5  ft.  are  how  many 
feet  ?  "    In  general,  such  problems  should  be  arranged  in  columns, 
as  is  done  in  ordinary  business  life ;  the  equation  form,  "  5  in. 
-f-  4  in.  =  9  in.,"  is  relatively  of  less  importance  and  should  come 
in  the  second  semester.     It  is  a  mistake,  for  various  reasons, 
to  attempt  to  treat  the  four  fundamental  processes  simultaneously. 
They  are  not  of  equal  difficulty ;  the  child  does  not  need  them  to 
an  equal  degree;  and  the  world  of  business  does  not  use  them 
with  equal  frequency.    Hence  addition,  the  easiest,  the  most  im- 
portant, and  the  most  interesting  to  the  child,  occupies  the  chief 
attention  in  this  year.     Incidentally,   as   needed   in   the   simple 
problems  proposed,  the  ideas  of  subtraction,  multiplication,  and 
division,  are  introduced ;  but  the  work,  even  in  addition,  is  limited 
to  the  number-space  I  to  20,  and  no  tables  are  learned. 

So  far  as  subtraction  is  treated,  the  "  making  change " 
method  should  be  used.  For  example,  "  If  you  have  10  cents, 
and  you  buy  a  pencil  costing  3  cents,  how  many  cents  have  you 


109]  Outline  for  the  First  Five  Grades  19 

left  ?  "  The  child  should  see  that  he  has  7  cents  left,  because 
3  cents  +  7  cents  =.  10  cents.  This  is  the  oral  subtraction  of 
business  life,  and  it  is  the  basis  for  the  "  Austrian  subtraction  " 
recommended  in  Grade  III. 

No  attempt  should  be  made  in  the  first  year  to  cover  system- 
atically all  number  relations  within  any  set  number-space. 

(/)  Symbols :  —  In  Grade  I  there  should  not  be  any  system- 
atic attempt  to  have  the  symbols  -}-,  — ,  X,  -j-  used  by  the 
children.  They  may  be  used  by  the  teacher,  and  explained,  and 
made  part  of  the  lessons  in  writing,  but  it  is  not  wise  to  give 
any  considerable  number  of  written  exercises  of  the  form  2  +  3 
—  4  +  i  =?  Still  more  objectionable  are  forms  like  2X3  +  2, 
and,  particularly,  2  +  2X3- 

The  reasons  for  opposing  such  examples  are  as  follows: 

(1)  These  chains  of  operations  enter  very  little  into  real 
mathematics.     In  practical   life  we  never  meet  a  problem  like 
2X3  +  4X6-^-3  —  2,   nor  do  we   find  these   symbols  much 
used  in  algebra  or  the  higher  mathematics.     Hence  they  should 
play  but  a  very  small  part,  if  any,  in  the  education  of  children. 

(2)  The  teacher's  personal  judgment  as  to  how  a  chain  of 
operations  ought  to  be  treated  has  no  validity  unless  supported 
by  the  conventions  of  mathematicians.     For  example,  a  teacher 
might  say  that  he  thinks  that  children  should  be  taught  that 
2  +  2  X  3  =  12,  taking  the  operations  in  the  order  stated;  but 
the  mathematical  convention  is  that  2  +  2X3  =  8,  the  multi- 
plication being  performed   first.     If  the  child   gets  the   wrong 
idea  now,  it  will  trouble  him  throughout  his  subsequent  mathe- 
matical work.     If  such  chains  are  to  be  given,  they  should  be 
in  forms  that  admit  of  no  misunderstanding,  as  2  X  3  +  2.     But 
even  these  are  open  to  such  serious  objection  that  they  should 
not  be  recommended. 

The  symbol  X  is  preferably  read  "  times  "  when  the  multi- 
plier comes  first  and  "  multiplied  by  "  when  it  comes  second,  as 

(a)  2  X  $3,  "  2  times  $3." 

(&)  $3X2,  "$3  multiplied  by  2." 

This  enables  the  sentence  to  be  read  from  the  left  to  the  right. 
The  reading 

(c)  $3X2,  "2  times  $3," 

has  good  authority,  particularly  in  England  and  France,  and 
among  older  American  writers,  but  it  is  coming  into  disfavor 


2O  Teachers  College  Record  [no 

because  it  is  not  in  accord  with  the  genius  of  our  language. 
The  reading  first  suggested  (a)  has  the  sanction  of  a  rapidly 
growing  number  of  our  best  writers,  and  is  thus  becoming  con- 
ventional. There  is  the  added  reason  for  it  that,  in  algebra,  the 
numerical  multiplier  is  generally  put  first,  as  in  ^ab.  Practically, 
however,  the  method  of  reading  the  X  is  not  so  serious  as  teach- 
ers often  suppose,  because  in  business  the  X  is  used  more  com- 
monly to  mean  "  by,"  as  in  the  expression  2  ft.  X  5  ft.,  3X9, 
where  it  does  not  indicate  multiplication.  Furthermore,  it  is  well 
to  know  that  in  mathematics  the  symbol  has  lost  all  standing, 
except  in  the  logical  statement  of  an  arithmetical  solution.  In 
algebra,  and  even  in  the  theory  of  numbers,  it  has  long  since 
been  nearly  forgotten,  the  dot  or  the  absence  of  sign  having 
taken  its  place,  as  in  2-4,  and  ^ab.  Indeed,  symbols  -)-,  — ,  X, 
-4-,  and  =,  were  invented  for  algebra  rather  than  for  arithmetic, 
and  their  chief  value  is  still  there,  although  the  X  and  -r-  have 
been  generally  discarded  by  algebraists.  It  is  only  in  the  written 
analysis  of  problems  in  the  grammar  school  grades  that  arith- 
metic has  much  use  for  them. 


3.  Problems  Suggestive  of  the  Type  Desired 

As  previously  stated,  arithmetic  is  merely  the  quantitative 
side  of  our  experience,  and  those  problems  will  be  of  most 
interest  that  are  drawn  from  fields  that  have  already  become 
attractive.  Hence  they  should  be  taken  from  other  school  studies, 
and  from  experiences  of  daily  life  outside  of  school.  Since  the 
curricula  of  the  Horace  Mann  and  the  Speyer  Schools  are 
radically  different  in  some  respects,  some  of  the  following  prob- 
lems that  are  suitable  for  the  one,  will  not  prove  fitting  for  the 
other  school. 

1.  With  toy  dishes,  or  blocks  to  represent  them,  set  a  table  for  four 
persons.     How  many  plates?  knives?  forks?  napkins?     How  many  spoons, 
with  two  for  each  person?     How  many  of  each  are  necessary  for  a  family 
of  four  persons?     For  your  family? 

2.  In  your  reader  find  the  following  pages  as  they  are  called:  5,  7,  21. 

3.  Count  by  s-cent  pieces ;  by  dimes. 

4.  Tell  the   length  and  breadth  of  each  of  these  blocks    (building 
blocks).      Name    each    by    its    measurement.      For    example,    2  X  4-inch 
block,  8-inch  block.     Use  these  names  always  in  building,  and,  in  general, 
call  denominate  numbers  by  their  full  names. 


in]  Outline  for  the  First  Five  Grades  21 

5.  Make  a  plan  of  a  room,  the  length  being  three  8-inch  blocks,  and 
the  width  two  8-inch  blocks.     Make  other  plans.     Measure  the  dimensions 
of  the  school  room,  of  the  yard.    Compare  the  school  room  in  size  with  the 
home  of  the  Pilgrims,  and  with  an  Eskimo  hut. 

6.  Show  a  foot-measure.     Draw  lines,  and  point  out  objects,  i  ft.  long. 
Do  the  same  with  the  yard  measure.    Estimate  the  length,  breadth,  and 
height  of  objects,  and  then  test  by  measurement.     For  example,  —  "Mary, 
estimate  John's  height;    now  measure  to  see  how  nearly  right  you  were, 
making  your  answer  correct  within  one  inch."     "  Give  what  you  think  to 
be  the  height  of  this  table,  and  measure  as  before."     Give  similar  problems 
for  chairs  and  doors.    Draw  the  plan  of  a  washcloth  of  suitable  size;   of 
an  iron  holder.     Give  their  dimensions. 

7.  Find  objects  that  you  think  weigh  I  Ib.     See  if  they  do.     Weigh 
various  objects  on  scales.     Likewise  use  other  measures  until  the  units 
of  measurement  are  quite  familiar. 

8.  A  good  milch   cow   averages  about   10  quarts   of  milk  per   day. 
Show  with  the  measures  how  much  that  would  be.     How  many  families 
could  she  supply  with  i  quart  for  each?    With  2  quarts? 

9.  Name  things  that  the  grocer  sells  by  the  quart,  the  pound,  the  bushel. 
Give  such  orders  as  your  mother  gives  to  grocers.     Learn  the  prices,  and 
pay  for  some  of  these  things  in  toy  or  real  money. 

10.  Do  the  same  in  connection  with  a  bakery  shop. 

11.  Using  sand  to  represent  sugar  and  other  commodities,  weigh  out 
what  different  children  call  for. 

12.  Make  measurements  on  paper  for  seed  envelopes,  boxes,  and  toy 
furniture  of  a  certain  size. 

13.  On  the  sand  table  lay  out  a  garden  according  to   some  scale ; 
include  the  walks  and  a  garden  plot  for  each  child.    Measure  the  growth  of 
plants  from  time  to  time. 

Problems  in  making  a  dirt  digger 

1.  The  stick  is  10  inches  long.    If  we  use  2  inches  for  the  point,  how 
much  is  left  for  the  handle  and  blade? 

2.  Half  of  the  remaining  part  is  for  the  handle ;  half  for  the  blade. 
What  is  the  length  of  each? 

3.  The  stick  is   ij4  inches  wide.     The  handle  is  to  be  half  of  that 
width.    How  wide  is  it? 

4.  How  much  must  be  cut  off  from  each  side,  in  order  to  leave  the 
handle  in  the  middle? 

Seed  marker 

Cut  strips  6  X  i  X  l/&  inch.  Draw  a  line  across  the  stick  il/2  inches 
from  the  end.  Find  the  center  of  the  end.  Draw  a  line  from  this  point 
to  each  end  of  the  cross  line.  Cut  the  stick. 

Wool  carders 

i.  Our  board  is  4  inches  wide.  How  many  rows  of  nails  will  it  hold 
*/2  inch  from  the  edge  and  Yz  inch  apart? 


22  Teachers  College  Record  [112 

2.  How  many  nails  are  needed  for  each  row,  when  the  board  is  4^ 
inches  long  and  the  nails  are  to  be  l/2  inch  from  the  edge  and  l/2  inch 
apart  ? 

Games 

1.  Draw  on  the  floor  or  table  three  concentric  circles  of  radii  3,  5, 
8  inches,  giving  each  circle  a  certain  value.     Now  roll  a  marble  so  as  to 
stop  it  in  one  of  these  circles,  and  count  the  value  given  in  favor  of  pupil.- 
Let  each  child  keep  his  own  score. 

2.  Bean-bag  game.     Have  a  board  with  a  hole  in  it,  and  three  bags 
of  different  sizes,  each  having  a  certain  value.    Throw  these  bags  through 
the  hole.    Let  each  child  keep  his  own  score. 

3.  Ring  toss.    Same  in  principle  as  the  preceding. 

4.  An  exercise  that  children  enjoy  very  much  is  counting  by  I,  2,  5, 
or  10,  while  certain  children  are  doing  a  given  task.    For  example,  see  if 
we  can  count  to  20  by  2's  while  the  pencils  are  being  collected.    Teachers 
find  this  to  be  a  valuable  drill. 

5.  Shuffle  board.     Make  bags  out  of  heavy  ticking.     Fill  them  with 
sand,  weighing  respectively  J4,  i,  il/2,  2  Ib.     Vary  from  time  to  time  the 
values  placed  upon  bags.    Have  the  captain  keep  the  score. 

Some  of  this  work  can  best  be  done  in  other  recitation 
periods  than  those  for  arithmetic,  even  though  a  separate  period 
for  arithmetic  is  planned.  At  such  times  it  is  important  to  keep 
in  mind,  as  a  pedagogical  principle,  that  any  one  recitation  should 
aim  primarily  to  do  work  of  one  kind  only,  be  it  literature,  learn- 
ing to  read,  nature  study,  or  something  else;  and  that  only  as 
much  apparently  foreign  subject  matter  should  be  allowed  as  is 
directly  necessary  for  the  accomplishment  of  this  one  aim.  If 
this  rule  is  not  followed,  a  recitation  period  may  be  occupied 
with  such  a  medley  of  matter  that  neither  children  nor  teacher 
can  tell  what  has  been  accomplished.  The  arithmetical  games 
suggested  should  soon  become  part  of  the  out-door  play. 

Grade  II 

i.  General  Suggestions 

(a)  Review  the  work  of  Grade  I  in  the  same  spirit  in  which 
it  is  there  taken.  In  general,  the  first  three  or  four  weeks  of 
each  year  should  be  devoted  to  a  review  of  the  work  of  the  pre- 
ceding year.  When  this  is  done,  the  common  complaint  of  the 
ignorance  of  children  of  the  preceding  work  may  cease. 

It  should  also  be  remembered  that  the  review  should  be  a 


113]  Outline  for  the  First  Five  Grades  23 

prominent  factor  in  every  lesson,  occupying  perhaps  half  the 
time  in  the  primary  grades.  Drills,  of  course,  are  included  under 
such  reviews.  Although  much  advance  has  been  made  in  edu- 
cation, drill  in  the  fundamental  operations,  even  to  the  point  of 
automatic  work,  has  not  yet  been  found  unnecessary.  In  the 
higher  grades,  it  is  less  necessary  to  plan  so  directly  for  reviews, 
because  they  are  incidental  to  the  solution  of  the  problems. 

Whatever  part  of  the  work  outlined  for  Grade  I  has  not 
been  covered  by  the  beginning  of  the  second  grade  in  the  Speyer 
School  will,  of  course,  be  taken  up  at  the  beginning  of  that  year. 
In  general,  there  is  likely  to  be  from  six  months  to  one  full 
year's  difference  between  the  pupils  of  the  two  schools  in  mathe- 
matical ability  and  knowledge.  This  outline  is  planned  primarily 
with  reference  to  the  Horace  Mann  School,  when  it  comes  to 
division  of  work  into  certain  grades. 

(&)  In  this  grade  the  work  in  arithmetic  comes  to  be  treated 
more  as  a  system,  the  child's  interest  having  now  been  aroused 
to  such  a  point  as  to  permit  it.  This  means  that  numerical  tables 
are  learned  as  such,  and  a  more  or  less  definite  amount  of  work 
is  laid  out  in  the  domain  of  pure  number.  As  in  other  studies, 
however,  this  abstract  part  should  follow  a  large  amount  of 
concrete  matter. 

(c)  There  is  no  radical  departure,  however,  from  the  kind 
of  work  done  in  Grade  I.    There  should  be  the  same  close  rela- 
tion to  manual  training,  to  other  studies,  and  to  the  problems  of 
daily  life.    The  children  should  be  encouraged  to  invent  problems ; 
the  quantities  of  food,  fuel,  and  clothing  used,  with  cost,  forming 
a  basis  for  much  of  the  work. 

(d)  Materials  as  before.    Also  cards  with  dots  and  figures. 
Dominoes   and  other  games,   including  the   number  games,   of 
the  Cincinnati  Game  Company.     Objective  work  in  establishing 
number  relations  is  still  necessary.     There  is,  for  example,  an 
advantage  in  counting  by  lo's,  and  in  seeing  bundles  of  splints. 
But  there  is  a  danger  in  carrying  the  objective  work  beyond  the 
needs  of  the  children,  as  there  is  in  abolishing  it  too  early.     A 
support  is  necessary  at  first  to  a  child  learning  to  walk ;  it  ceases 
to  be  necessary  later;  to  tell  when  it  ceases  to  be  necessary  and 
begins  to  be  harmful,  demands  careful  thought  for  each  indi- 
vidual, —  and  so  for  objective  teaching.     But  the  children  them- 
selves may  be  relied  upon  for  some  aid  in  this  matter,  if  a  pride 


24  Teachers  College  Record  [114 

has  been  developed  in  them  to  abandon  the  use  of  objects  as 
soon  as  possible. 

(e)  If  the  oral  work  is  rapid  and  accurate,  the  written  work 
will  be  so.  But  in  quantity  the  oral  work  should  greatly  pre- 
dominate. There  is  a  temptation  to  have  too  much  written  arith- 
metic at  this  stage,  simply  because  it  is  easily  assigned  for  seat 
work.  The  great  danger  is  that  such  young  children,  thus  left 
alone  at  their  seats,  will  drop  into  careless  habits  involving  divi- 
sion of  attention.  No  text-book  is  advisable,  at  least  in  the  first 
part  of  this  year,  for  reasons  previously  given. 

(/)  In  the  written  work  the  same  attention  should  be  given 
to  neatness  and  accuracy  that  is  required  in  other  written  exer- 
cises. Indeed,  these  qualities  are  particularly  important  objects 
of  written  arithmetical  work  in  this  grade. 

The  equation  form  may  now  be  introduced ;  as  24-5=?, 
2+?  =  7,  ?  +  8  =  8,  2X?  =  6,  6  +  ?  =  2;  or  with  n  (for 
number)  in  place  of  the  "  ?."  But  it  should  be  remembered  that 
these  are  not  the  forms  of  practical  life,  and  more  time  should 
be  directed  to  the  forms  more  common;  namely,  the  work  in 
columns.  Such  symbols  of  operation  are  necessary  later,  when 
the  child  conies  to  the  explanation  of  problems  of  some  length; 
in  this  grade  they  enter  only  incidentally.  If  the  teacher  gives 
chains  of  operations  like  2  +  3  —  1+2,  or  2X3  —  4,  mathe- 
matical conventionalities  must  be  observed ;  e.g.,  2  -j-  3  X  4  = 
14,  not  20,  and  hence  it  would  be  better  to  write  this  3X4  +  2, 
so  that  the  operations  could  be  taken  in  their  order.  (See  p.  19.) 
Written  work  of  this  kind  is  not,  however,  very  valuable;  the 
oral  treatment  is  better. 

(g)  Bibliography :  —  Smith :  Teaching  of  Elementary  Math- 
ematics, Chaps,  i-v ;  McClellan  and  Dewey :  Psychology  of  Number, 
Chaps,  i-ix;  Phillips,  in  the  Pedagogical  Seminary,  Oct.,  1897. 

2.  The  Mathematical  Work 

(a)  The  number-space,  i  to  10,000  for  reading  (as  in  larger 
street  numbers),  I  to  1,000  for  counting  and  writing,  I  to  100 
for  operations.  The  Roman  numerals  may  be  limited,  as  in 
Grade  I,  to  the  space  I  to  XII,  this  being  sufficient  for  the  present 
uses  of  the  children;  although  many  city  courses  carry  this 
system  to  C. 


115]  Outline  for  the  First  Five  Grades  25 

(&)  Counting:  —  The  counting  of  Grade  I  is  carried  for- 
ward. Counting  by  twos,  from  2  to  20 ;  by  threes,  from  3  to  30 ; 
by  fours,  from  4  to  40 ;  the  object  being  to  lay  the  foundation  for 
the  multiplication  table  to  4  X  10.  This  counting  will  naturally 
and  incidentally  extend  farther  as  an  interesting  exercise. 

Also  counting  by  twos  from  i  to  9,  by  threes  from  I  and  2 
to  10,  and  by  fours  from  i,  2,  3,  to  n  ;  the  object  being  to  form 
the  tables  of  addition.  This  work  can  easily  be  made  an  interest- 
ing exercise  in  games,  and  rapid  oral  work  will  have  an  interest 
per  se.  But  for  interest,  much  dependence  should  be  placed  on 
the  proper  kind  of  problems.  The  meaning  of  "  dozen." 

(c)  Tables:  —  When   the   counting   suggested   in    (&)    has 
been  carried  out,  the  children  know  the  table  of  addition,  and 
also   the   table   of   multiplication    through    4  X  10.      The    latter 
should,  however,  be  learned  and  drilled  upon  as  such.     Also  the 
table  of  ID'S  and  5's,  the  former  to  100,  the  latter  to  50.     The 
subtraction  table  need  not  be  developed  and  learned,  since  sub- 
traction is  made  to  depend  upon  addition  as  stated  below. 

(d)  Measuring:  —  The  measures  chiefly  used  in  this  grade 
are  those  used  in  Grade  I,  with  a  few  additions.    They  are: 

Length,  —  the  inch,  foot,  yard. 
Capacity,  —  the  gill,  pint,  quart,  gallon ; 

the  quart,  peck,  bushel; 

the  cubic  inch. 

Weight,  —  the  ounce,  pound. 
Time,  —  the  day,  week,  month; 

minutes  in  hour,  hours  in  day,  days  in  week. 
Surface,  —  the  square  inch,  square  foot. 
Here,  as  in  all  grades,  the  children  should  become  thoroughly 
familiar  with  the  measures  by  actual  use. 

(e)  Operations:  —  Addition  of  several  one-figure  num-       2 
bers ;  or  of  two- figure  numbers,  neither  column  at  first  ex-       3 
ceeding  9.    Accustom  the  children  to  say,  "  5,  14,  21,  24,  26,"       7 
from  below;  and  also,  "2,  5,  12,  21,  26,"  from  above.     In       9 
general  the  child  should  read  a  column  as  he  reads  a  sentence,       5 
never  counting  by  units  any  more  than  he  would  spell  by    — 
letters.  26 

Subtraction  has  from  the  earliest  times  been  taught  in 
various  ways.  Some  methods  of  treatment  are  more  rapidly  per- 
formed, others  are  more  easily  explained.  Since,  however,  the 


26  Teachers  College  Record  [116 

child  is  not  expected  to  learn  or  to  repeat  any  but  very  simple 
explanations,  and  since  the  purpose  of  an  explanation  is  to  justify 
a  process  that  soon  should  be  merely  mechanical,  it  is  reasonable 
to  expect  that  the  shortest  operation  is  the  one  that,  in  the  long 
run,  will  be  accepted  by  the  world.  This  is  seen  in  many  of  the 
other  operations.  For  example,  in  the  division  of  fractions,  the 
world  now  inverts  the  divisor  and  multiplies,  although  for 
generations  it  divided  by  reducing  the  fractions  to  a  common 
denominator. 

Multiplication  is  in  this  year  limited  to  one-figure  factors, 
and  to  the  domain  I  X  i  to  4  X  10. 

Division  is  limited  chiefly  to  "  exact  division,"  not  involving 
mental  "  carrying."  It  is  introduced  only  as  needed,  in  simple  and 
natural  problems,  as  in  taking  y2,  Y$,  etc. 

The  terms  addend,  sum,  subtrahend,  minuend,  difference, 
multiplier,  are  used  as  necessary,  but  no  formal  definitions  are 
given. 

Fractions:  Halves,  thirds,  fourths,  fifths,  sixths,  eighths, 
tenths.  The  seventh  and  ninth  are  omitted,  or  are  met  only 
incidentally,  there  being  almost  no  use  for  them  in  the  measure- 
ments and  the  problems  of  this  grade.  In  general,  no  inexact 
fractional  parts,  like  y$  of  7;  such  cases  may,  however,  enter 
into  certain  impromptu  problems. 

3-  Suggestive  Problems 

i.  Practice  stepping  off  2  feet  at  each  step.  Then,  by  stepping,  find 
the  width  of  the  street  in  front  of  your  school-building.  What  is  the 
width  of  the  widest  streets  near  you  ?  What  seems  to  be  the  average  width 
of  streets  in  this  city?  In  the  same  way  find  the  width  and  depth  of 
near-by  lots.  Show  the  size  of  an  ordinary  yard  in  a  village. 

2.  How  many  children  can  be  accommodated  at  the  blackboard  in 
your  school-room  at  one  time,  counting  2^2  feet  for  each  child  ? 

3.  Measure  your  school-room.     Make  a  floor  plan  of  it,  with  blocks, 
according  to  some  scale.    Likewise  measure  and  make  a  floor  plan  of  some 
grocery  store  you  have  visited;  also  of  a  bakery. 

4.  Name  the  months  in  each  of  the  four  seasons.     Each  season  is 
what  part  of  one  year? 

5.  Measure  the  growth  of  plants,  twigs,  etc.,  correct  to  within  ^  inch. 

6.  Make  a  floor  plan  of  the  apartment,  or  of  one  of  the  floors  of  the 
house,  in  which  you  live.    Use  blocks  or  make  a  drawing  to  scale. 

7.  How  many  chairs  are  needed  for  a  six-room  apartment  containing 
4  persons?    Explain  your  answer. 


117]  Outline  for  the  First  Five  Grades  27 

8.  Give  the  size  of  a  fair-sized  napkin.    Of  a  towel.    Show  by  draw- 
ings.   How  many  towels  are  needed  to  supply  a  family  of  four?    Explain 
your  answer. 

9.  Read  the  thermometer   from  time  to  time  and  explain   what  it 
signifies. 

10.  Count  the  number  of  grains  on  an  ear  of  corn.    One  grain  of  corn 
often  produces  a  stalk  with  three  such  ears. 

11.  A  family  in  this  city,  burning  coal  in  the  range,  often  spends 
25  cents  per  week  for  kindling,  receiving  3  bundles  of  sticks  for  5  cents. 
What  is  the  cost  per  month?    How  many  bundles  are  burned  per  week? 

12.  It  takes  zy2  gallons  of  milk  to  make  I  Ib.  of  butter.    One  gallon 
weighs  9  Ibs.     How  many  pounds  of  milk  make  i  Ib.  of  butter?     Milk 
costs  us  about  7  cents  per  quart.    What  is  the  worth  of  the  milk  necessary 
for  I  Ib.  of  butter?    What  is  the  cost  of  one  Ib.  of  butter?    How  does  this 
compare  with  the  cost  of  the  milk?    Why  this  difference? 

13.  Give  the  dimensions  of  a  pencil-box  of  suitable  size  for  use. 

14.  Games,  including  dominoes,  and  the  number  games  of  the  Cin- 
cinnati Game  Company. 


Grade  III 

i.  General  Suggestions 

(a)  Review  the  work  of  Grade  II,  devoting  three  or  four 
weeks  to  it.    See  General  Suggestions  (a)  under  Grades  I  and  II. 

(b)  The  text-book  is  necessary  in  this  grade.     Some  reasons 
for  the  delay  have  already  been  given.     (See  page  16.)     The  use 
of  the  text-book  is  that  it  may  serve :  —  ( i )  as  a  reference  book 
for  facts,  like  a  dictionary,  as  in  the  case  of  tables;   (2)  as  a 
collection  of  problems  to  save  the  time  of  dictation.     The  prob- 
lems often  fail  to  appeal  to  interest  and  must  be  supplemented 
or  omitted,  but  the  choice  of  evils  leads  to  using  a  book  from 
now  on.     There  is  a  serious  danger,  however,  in  introducing  a 
book,  of  departing  too  radically  from  the  child's  daily  interests. 

(c)  The  work  in  this  grade  becomes  still  more  systematic. 
The  object  of  the  year,  so  far  as  operations  are  concerned,  is  to 
become  familiar  with  "  the  four  processes  "  in  the  number-space 
i  to  i  ,000,  division  being  limited  as  set  forth  below.    The  object, 
so  far  as  rich  thought-content  is  concerned,  is  to  acquaint  the 
pupils  with  numerous  separate  facts  of  interest,  and  thus  to  reveal 
to  them  the  quantitative  side  of  a  few  large  topics. 

(d)  Objective  work  becomes  less  and  less  needed  in  number 
relations  of  integers.     But  it  is  just  as  necessary  as  ever  in 


28  Teachers  College  Record  [118 

presenting  radically  new  subjects,  as  in  reducing  fractions,  in 
studying  new  forms,  and  in  taking  up  new  measures. 

(e)  Written  work  is  still  subordinate,  being  confined  largely 
to  computations  too  extensive  for  purely  mental  treatment.  A 
large  amount  of  rapid  oral  analysis  should  be  given  in  very 
abbreviated  form.  Elaborate  analysis  should  be  avoided  both 
by  the  teacher  and  the  pupil. 

(/)  There  is  a  great  advantage  in  the  occasional  use  of  oral 
analysis,  giving  the  main  steps  without  performing  the  operations. 
For  example:  If  a  book  costs  $2,  how  much  will  36  such  books 
cost  ?  Answer :  "  36  books  will  cost  36  times  $2." 

Since  children  often  fail  to  get  the  exact  condition  of  a 
problem,  it  is  well  to  have  them  restate  it,  in  their  own  words, 
if  it  admits  of  such  a  change.  It  is  well  also  to  have  them  state 
the  number  of  steps  involved  in  the  solution,  showing  very  briefly 
what  they  are,  and  giving  the  approximate  answer.  The  actual 
solution  may  then  follow,  or  may  be  omitted  entirely  for  the 
time.  Such  work  forces  children  into  the  "  thought-side "  of 
arithmetic,  and  away  from  the  impression  that  it  is  all  "  mere 
figuring" ;  in  other  words,  it  overcomes  the  tendency  to  become  too 
formal.  The  oral  solution  of  problems  with  only  approximate 
answers  may  well  increase  in  prominence  as  the  work  advances. 
One  argument  in  its  support  is  the  fact  that  many  of  the  problems 
of  adults  are  actually  solved  only  approximately;  also,  a  very 
large  percentage  of  adults  use  the  approximate  answer  as  a  guide 
to  correctness  in  the  actual  calculation. 

(g)  Geographical  considerations  should  influence  the  subject 
here  as  in  the  other  grades.  (See  Grade  I,  2,  (c).)  For 
example,  such  words  as  penny,  bit,  shilling,  nickel,  are  common 
as  names  of  coins,  in  some  parts  of  the  country,  but  not  in  others. 

(h)  It  is  important  to  consider  how  far  induction  is  legiti- 
mate in  elementary  arithmetic.  The  world  induced  before  it 
deduced;  and  the  child  does  the  same.  Even  in  the  highest 
mathematics  induction  plays  a  great  part,  in  discovery;  then 
deduction  comes  to  play  its  part  in  proving  that  the  induction  is 
correct.  So  in  the  primary  grades  induction  is  not  only  allow- 
able, but  it  is  the  natural  method.  The  child  comes,  however, 
in  the  later  grades,  to  the  period  of  more  or  less  rigid  deduction, 
because  he  is  largely  applying  principles  already  familiar.  Then, 
however,  another  kind  of  induction  should  be  going  on,  since 


119]  Outline  for  the  First  Five  Grades  29 

(see  page  50)  he  should  be  reaching  new  generalizations  about 
industries  and  other  phases  of  our  quantitative  experience.  Truly 
inductive  work  in  arithmetic  requires  that  the  rule  be  put  in  the 
background  at  first;  concrete  problems  should  first  occupy  the 
attention,  and  only  after  several  of  these  have  been  solved,  and 
the  methods  compared,  should  the  rule  itself  be  broached  and 
worded.  Even  then  there  is  a  great  danger  in  making  altogether 
too  much  of  memorizing  rules,  particularly  those  of  operation. 
Teachers  should,  however,  distinguish  between  a  rule  that  is 
originated  by  the  pupil,  and  one  which  is  dogmatically  given  to 
him.  The  former  has  high  value;  the  latter  is  dangerous. 

For  example,  take  this  problem:  Your  garden  is  24  yards 
long  and  3  yards  wide.  How  many  feet  of  wire  will  it  take  to 
enclose  it?  A  rectangle  represents  the  plot  of  ground,  and  we 
have  this  column  for  addition:  24 

24 

3 

3 

Different  devices  may  be  employed  to  explain  the  process, 
as  shown  on  page  31. 

Another  example  might  be :  A  soldier  on  a  long  march  must 
carry  a  knapsack  weighing  2  Ibs;  food  in  it  weighing  4  Ibs; 
a  gun  weighing  9  Ibs ;  and  a  blanket  weighing  14  Ibs.  How 
much  must  he  carry  in  all? 

Similar  problems  should  be  solved  before  an  attempt  is  made, 
if  ever,  to  phrase  any  rule  of  addition. 

The  following  might  be  suitable  problems  in  subtraction :  — 
(i)  A  good  carriage  horse  often  weighs  1,025  Ibs.  A  good 
dray  horse  often  weighs  1,850  Ibs.  What  is  the  number  of 
pounds  of  difference?  Why  need  there  be  such  a  difference 
in  weight?  (2)  In  May,  1902,  anthracite  coal  sold  at  $5.25 
per  ton;  in  August,  owing  to  the  strike,  at  $10.00  per  ton. 
How  much  advance  was  made  in  price?  (3)  For  children  who 
have  been  studying  history:  How  many  years  is  it  since  Henry 
Hudson  sailed  up  the  Hudson  River? 

In  truly  inductive  work  there  is  no  feeling  of  hurry  to  reach 
the  rule ;  the  latter  is  gradually  brought  to  light  through  numerous 
positively  concrete  examples. 

(i)   Types :  —  The  question  arises,  How  should  the  rule  be 


3O  Teachers  College  Record  [120 

recalled  by  the  child  ?  Shall  he  learn  the  teacher's  rule  verbatim  ? 
Or  shall  he  memorize  a  wording  of  his  own?  Or  shall  he  recall 
some  practical  problem,  and  reproduce  the  process  there  fol- 
lowed? Many  teachers  prefer  the  last  plan.  For  example,  the 
garden  mentioned  on  page  29  is  24  yards  long  and  3  yards  wide. 

20+4  24        24 

20  -j-    4  24        24 

3  or    3  or    3 

>      3  33 


40  +  14  =  54        14        54 
40 

54 
In  subtraction,  taking  the  example  about  the  horses,  we  have: 

1,850     1,850  =  i, 800  -(-40+10 
less  1,025     1,025=1,000+20+    5 


800  +  20+    5  = 

For  a  discussion  of  the  various  methods  of  solution  in  sub- 
traction see  page  31.  Whatever  the  method  followed,  the  par- 
ticular example  is  held  in  memory  for  some  time  as  an  aid  to 
the  solution  of  new  problems.  Being  concrete,  it  can  easily  be 
remembered;  and  since  it  must  be  recalled  many  times  for  help, 
the  rule  itself  gradually  comes  to  light  and  may  finally  be  care- 
fully worded,  —  preferably  in  the  pupil's  own  way.  This  plan 
avoids  the  premature  learning  of  the  generalization. 

(/)  Bibliography :  —  Smith :  Teaching  of  Elementary  Mathe- 
matics; and  McClellan  and  Dewey:  Psychology  of  Number,  as 
before ;  McMurry :  Method  of  Recitation,  Chap.  2,  p.  14. 

2.  The  Mathematical  Work 

(a)  The  number-space: — i  to  1,000,000  for  reading  and 
writing,  i  to  1,000  for  operations.  The  Roman  numerals  I  to  D, 
but  not  to  work  out  every  numeral,  the  reading  of  chapter  num- 
bers being  the  important  feature. 

(&)  Counting:  —  The  counting  of  Grades  I  and  II  is  con- 
tinued sufficiently  to  complete  the  development  of  the  multiplica- 
tion table  to  10  X  10,  which  is  as  far  as  it  need  be  carried. 


I2i  ]  Outline  for  the  First  Five  Grades  31 

(c)  Tables  of  abstract  number:  —  The  multiplication  table 
as  above,  thoroughly  learned  and  made  the  object  of  continued 
and  rapid  oral  drill.    The  addition  table  reviewed  daily,  and  also 
made  the  object  of  continued  oral  drill. 

(d)  Measuring:  —  In  addition  to  the  measures  introduced 
in  Grades  I  and  II,  the  following  are  studied : 

Length,  —  the  rod  and  mile.     Number  of  New  York  blocks 
to  the  mile,  north  and  south,  also  east  and  west. 
Weight,  —  the  ton. 
Time,  —  months  in  year ; 

days   in   months,   "  Thirty   days  hath   September," 

etc.; 

days  in  year; 

weeks  (approximately)  in  the  month  and  year. 
Meaning  of  "  quire  "  and  "  score." 

Particular  attention  to  the  table  of  United  States  money,  but 
always  omitting  the  "  eagle,"  and  omitting  the  "  mill "  in  the 
primary  grades. 

(e)  Operations:  —  the   four   fundamental   processes   in  the 
number-space   I   to  1,000,  division  being  limited  to  divisors  of 
two  figures. 

In  addition,  teachers  are  not  limited  to  any  one  device.  In 
explaining  a  process  involving  some  mental  retention  of  number, 
as  in  "  carrying,"  it  may  be  necessary  to  consider  sticks  in  bundles 
of  10,  as  is  often  done,  and  to  adopt  also  such  devices  as 

26  =  20  +    6  26 

39  =  30+9  39 

50+15-65     15 
50 

65 

in  order  to  bring  out  the  idea.  Since  no  explanation  should  be 
required  from  the  child  at  this  stage,  further  than  the  answer 
to  simple  questions,  all  that  is  necessary  is  that  he  should  (i) 
know  how  to  perform  the  operation,  and  (2)  feel  that  he  has 
taken  nothing  on  mere  authority,  beyond  the  conventionalities  of 
the  subject. 

In  subtraction,  of  the  various  methods  suggested,  the  one 
that  is  now  having  the  greatest  favor  is  the  so-called  "  Austrian 


32  Teachers  College  Record  [122 

method,"  which  has  the  advantages  of  brevity  and  of  requiring 
no  subtraction  table  to  be  memorized.  The  general  steps  neces- 
sary to  its  development  are  : 

(  i  )  Meaning  of  difference,  —  that  number  which  added  to  the 
smaller  number  produces  the  larger,  —  an  idea  sufficiently  com- 
plete for  children.  For  example  ;  9  —  5  =  4  because  5  -f-  4  =  9. 
By  recalling  the  addition  table,  such  differences  are  at  once  49 
brought  to  mind.  Apply  to  cases  like  49  —  25  :  25 

Any  subtraction  of  two-place  numbers  where  no  figure  — 
in  the  subtrahend  signifies  a  larger  amount  than  the  corre-  24 
sponding  figure  in  the  minuend,  follows  this  plan. 

(2)  Adding    10,    100,   or   1,000   to   each   number   does   not 
change  the  difference.    Inductively  shown  from 

9     19     29 

5     15    25 

444 

(3)  Consider  the  case  of  52. 

27 


. 
Excluding  the  advanced  idea  of  negative  numbers  from  this 

discussion,  we  may  say,  there  is  no  number  which  added  .  to  7 
makes  2,  but  5  added  to  7  makes  12.  We  have  now  increased 
the  52  by  10,  and  we  must  add  10  to  27  so  as  not  to  change  the 
difference.  3  (tens)  and  2  (tens)  are  5  (tens).  Hence  the  dif- 
ference is  25. 

Like  all  innovations  in  our  own  long  established  mental  pro- 
cesses, this  may  seem  difficult;  but  looked  at  from  the  stand- 
point of  the  child  it  saves  memorizing  many  facts,  and  it  reduces 
subtraction  to  the  much  simpler  operation  of  addition.  The  ob- 
jection that  it  does  not  give  the  notion  of  "  taking  away,"  is 
hardly  worth  notice;  it  gives  the  result  with  a  minimum  of 
mental  friction,  and  this  is  the  practical  end  in  view.  The  details 
of  presentation  are,  of  course,  capable  of  great  variation,  in- 
volving the  use  of  materials,  and  such  devices  as  : 

52  ==  50  -\-  2      the  difference  being  the  same      50+12 
27  —  20  -f-  7     as  between  30  -f-    7 

20+5 


123] 


Outline  for  the  First  Five  Grades 


33 


In  short  division,  it  may  frequently  be  necessary  to  use  the 
"  long  division  "  form,  showing  that  the  former  is  only  an  abbre- 
viation of  the  latter.  And,  in  general,  the  abridgment  naturally 
follows  the  more  complete  form  in  all  operations.  As  in  addi- 
tion, an  introduction  to  "  carrying  "  often  requires  such  a  device 
as  this : 

27 

36 

13 
50 

63 

and  as  in  subtraction  we  have  to  resort  to  complete  forms  like  this : 
The  difference  between  52  and  27  equals  that  of 

40  -f-  12    or    50  -f-  12 
and  20+7  and  30  -f-    7 

20+5 
so  in  multiplication  we  lead  to 

12  12 

7    through      7 


84 


and  in  division  we  lead  to 
5)134 


14 
70 

84 


through 


26,  and  4  remainder, 
26 


5)134 
100 


34 
30 


4  remainder, 
or  through       5 1 100  4-30 

"-  * 


I        \J~       I 

20  -j-    6,  and  4  remainder. 


34  Teachers  College  Record  [124 

No  definite  rule  can  be  laid  down  for  these  cases.  The 
teacher  should  use  the  more  complete  form  whenever  necessary 
for  the  class  in  hand,  and  come  to  the  abridged  form  as  soon 
as  the  pupils  are  ready  for  it. 

We  shall  find  that  it  is  generally  better  in  "  long  division  " 
to  put  the  quotient  above  the  dividend.  (See  page  39.)  In  "  short 
division  "  this  could  also  be  done  were  it  worth  while  to  change 
the  well  established  custom  of  writing  it  underneath. 

(/)  Geometric  forms.  The  square  and  rectangle,  as  met 
in  manual  training,  basketry,  paper-work,  and  elsewhere,  are 
studied  with  reference  to  perimeter  and  area.  The  cubic  inch 
and  cubic  foot.  Study  of  the  circle,  including  the  diameter  and 
the  radius,  but  without  mensuration.  The  details  of  the  work  in 
geometry  are  reserved  for  a  subsequent  article. 

(g)  Fractions  and  ratios  are  incidentally  studied,  carrying  on 
the  work  of  Grade  II.  There  is  no  formal  treatment  of  the  sub- 
ject. Such  simple  reductions  as  f  =  £,  f  —  i>  and  such  operations 
as  £  +  i,  \  of  £,  \  —  \  are  made  part  of  the  oral  work. 

The  decimal  form  is  used  in  writing  United  States  money, 
but  without  any  explanation  of  the  decimal  system.  For  the  sake 
of  practice,  papers  may  be  marked  in  per  cent,  and  the  term  should 
be  explained  and  used  in  other  work. 


3.    Suggestive  Problems 

1.  One  gas  jet  burns  about  5  cubic  feet  of  gas  per  hour.    How  much 
gas  is  consumed  per  week  in  a  house  that  burns  3  jets  each  evening  from 
7  to  10.30  o'clock?     Estimate  the  amount  that  a  street  lamp  burns  in  one 
month. 

2.  Gas  costs  $i  per  thousand  cubic  feet.     Estimate  the  cost  of  the 
different  amounts  of  gas  above  mentioned.     Estimate  the  cost  of  gas  for 
street  lights  for  a  distance  of  one-quarter  of  a  mile  on  one  street. 

3.  A  faucet  actually  leaked  in  a  New  York  house  at  the  rate  of  i  pint 
of  water  every  thirty  seconds.    How  much  would  that  amount  to  in  one  day 
of  twenty- four  hours?     In  one  month?     The  water  was  allowed  to  run 
six  months;  how  much  was  the  waste? 

4.  Make  out  an  actual  bill  for  ice  for  your  family  for  one  month. 

5.  Live  cattle   sell   at  7l/2   cents  per  Ib.    Good  beef-steak  now  costs 
28  cents  per  Ib.    What  is  the  difference?    Why  is  this  difference? 

6.  Make  yourself  acquainted  with  some  of  the  more  common  fish, 
and  the  average  weight  of  each.  Tell  what  one  fish  of  each  kind  will  cost 
at  the  customary  price  per  pound. 


125]  Outline  for  the  First  Five  Grades  35 

7.  Estimate  the  number  of  persons  living  in  some  apartment  house 
near  you,  and  explain  how  you  do  it. 

8.  Tell  time  by  the  clock,  to  minutes  and  fractions  of  a  minute. 

9.  What  distance  do  you  walk  each  day  in  going  to  and  from  school 
once?     Show  how  you  find  it. 

10.  In  order  to  run  a  single  ordinary  rock  drill,  used  for  blasting,  one 
engineer  and  two  men  are  necessary.    The  leading  man  with  the  drill  often 
receives  $2.75  per  day;  his  assistant,  $1.50;  the  engineer,  $3.    One  drill  will 
keep  a  blacksmith  busy  about  one-fourth  of  the  time,  sharpening  the  drills ; 
he  receives  $3  per  day.    How  much  does  the  labor  cost  to  run  a  single  drill 
one  day? 

11.  A  business  man  often   sends    18  letters  per  day,  6  days  in  the 
week,  each  requiring  a  2-cent  stamp.    At  that  rate,  what  is  his  postage  bill 
per  month  of  four  weeks? 

12.  Talk  with  a  gardener  to  find  out  how  large  a  lettuce  bed  it  would 
take  to  supply  all  the  lettuce  your  family  would  need  for  a  season.    Show 
how  you  estimate  it. 

13.  Young  chickens  are  now  worth  20  cents  per  Ib.    What  is  the  cost 
of  one  weighing  2^2  Ibs. ?     Cost  per  dozen? 

14.  The  chicken  industry : — 

(a)  Of  the  eggs  that  a  farmer  "sets"  for  raising  chickens,  seldom 
more  than  4  out  of  5  hatch  out.  If  he  sets  600,  how  many  chickens  does 
he  get? 

(fe)  Of  those  that  are  hatched  out,  about  i  out  of  3  is  drowned,  killed 
by  wild  animals,  or  sickens  and  dies,  before  the  age  of  3  months  is  reached. 
How  many  grow  up  from  the  600  eggs? 

It  would  be  well  to  continue  a  story  like  this,  referring  to  the  number 
eaten  by  a  family,  the  number  sold,  the  weight  and  price  per  pound,  and 
the  amount  and  cost  of  feed.  Similar  series  of  problems  in  raising  a  herd 
of  sheep.  Estimate  the  cost  of  feeding  a  horse  in  this  city. 

15.  A  motorman  on  a  New  York  trolley  car  very  frequently  runs  his 
car  80  miles  per  day.     How  many  times  does  he  stop  if  he  averages  one 
stop  every  third  block,  counting  20  blocks  to  one  mile? 

16.  The  conductor  often  collects  75  fares  on  one  trip  "  up  town," 
covering  about  n  miles.     How  many  dollars  would  this  amount  to,  count- 
ing fares  at  5  cents  each? 


Grade  IV 

i.  General  Suggestions 

(a)  Review  the  work  of  Grade  III,  devoting  three  or  four 
weeks  to  it.  See  General  Suggestion  (a)  under  Grades  I  and  II. 

(&)  On  the  use  of  the  text-book,  see  General  Suggestion  (&) 
under  Grade  III. 


36  Teachers  College  Record  [126 

(c)  Touching  processes,  the  work  of  the  year  is  chiefly  given 
to  the  ability  to  handle  the  four  operations  in  general  problems 
and  to  the  addition  and  subtraction  of  common  fractions,  the  essen- 
tially new  work  being  the  completion  of  "  long  division."    As  to 
quantitative  facts  of  value,  it  is  now  possible  to  present  the  prin- 
cipal sides  of  many  leading  industries,  and  thus  to  build  up  as 
full  and  vivid  a  picture  of  many  matters  as  is  done  in  other 
studies,    though    always    through    figures    rather    than    through 
words. 

(d)  This  is  the  grade  in  which  children  often  begin  to  lose 
interest  in  the  subject,  and  well  they  may  if  processes  only  are 
depended  upon  to  arouse  life.    It  is  the  teacher's  duty  to  plan  much 
variety  in  method,  and  to  make  fair  use  of  competition ;  but  more 
than  this  should  be  done.    Really  valuable  subjects  should  be  se- 
lected; interest  from  any  other  source  is  likely  to  be  largely  arti- 
ficial.    The  text-book,  of  course,  contains  but  few  such  topics, 
and  the  teacher  will  simply  have  to  do  the  best  she  can,  under 
present  circumstances,  to  find  them  herself. 

(e)  On  the  proper  use  of  diagrams.     These  can  be  exten- 
sively used  to  advantage,  and  should  be  drawn  and  studied  before 
the  computation  begins ;  otherwise,  the  child  wanders  about  in  the 
dark  and  gets  no  good  from  them.    For  example :  —  In  finding  the 
area  of  a  floor,  the  diagram,  if  needed  at  all,  is  needed  before  the 
computation.    In  general,  the  children  should  become  accustomed 
to  thinking  out  the  main  steps  to  be  taken  and  the  approximate 
answer,  before  beginning  the  computation. 

(/)   For  further  suggestions,  see  Grades  I,  II,  III. 
(g)  Bibliography :  —  Smith,    McClelland   and   Dewey,   and 
McMurry,  as  noted  in  the  preceding  grades. 

2.  The  Mathematical  Work 

(a)  The  number  space.  No  limits  need  be  observed  hereafter, 
except  that  number  names  beyond  billions  are  of  little  value. 
Roman  numerals  up  to  MM  being  sufficient  for  dates,  form  the 
limit  for  this  system  in  the  grades. 

(&)  Tables.  The  addition  and  multiplication  tables  have 
already  been  learned,  and  are  to  be  made  the  subject  of  constant 
review. 

The  following  table  of  aliquot  parts  should  become  known 


127]  Outline  for  the  First  Five  Grades  37 

this  year,  in  anticipation  of  the  work  in  decimal  fractions  for 
next  year: 

y2  =  0.50  =  $0% 

y4  =  0.25  =  25% 


=  0.2O  =  20% 

=  O.IO==  10% 


Such  facts,  like  the  rules  of  arithmetic,  should  be  reached 
through  study  of  concrete  examples.  The  other  plan,  deductive 
as  it  is  called,  though  not  truly  deductive,  is  to  study  each  fact 
and  the  rules  first  ;  and,  after  having  learned  them,  to  apply  them 
to  examples. 

(c)  Measures.  This  year  should  see  the  tables  of  denominate 
numbers  systematized,  and  the  great  basal  units  of  arithmetic 
emphasized. 

The  following  tables  are  given  as  the  ones  to  be  memorized, 
the  chief  units  being  italicized  : 

Length.  12  in.  (12")  =  i  //.  (:') 

3  ft.  =  i  yd. 

Sl/2  yds.  or  i6l/2  ft.  =  i  rd. 
320  rds.  or  5280  ft.  =  i  mi. 

The  surveyor's  table  is  omitted,  as  being  part  of  the  technical 
education  of  a  surveyor,  and  unnecessary  in  common  life. 
Surface.  144  sq.  in.  =  i  sq.  ft. 

9  sq.  ft.  =  i  sq.  yd. 
30^4  sq.  yds.  =  i  sq.  rd. 
1  60  sq.  rds.  =  i  acre 
640  acres  =  i  sq.  mi. 

In  certain  communities  the  study  of  township  sections  would 
have  place. 
Capacity. 

Liquid.  Dry.  Cubic. 

4  gi.  =  i  pt.  2  pts.  =  i  qt.  1728  en.  in.  =  i  cu.  ft. 

2  pts.  =  i  qt.  8  qts.  =  i  pk.  27  cu.  ft.  =  i  cu.  yd. 

4  qts.  =  i  gal.  4  pks.  =  i  bu.  A  "cord  of  wood"  ex- 

plained. 
Weight.  1  6  oz.  =  i  Ib. 

2000  Ibs.  =  i  ton 


38  Teachers  College  Record  [128 

This  table  of  avoirdupois  weight  is  the  one  which  the  children 
will  practically  need.  The  long  ton  should  be  mentioned,  but  troy 
weight  and  apothecary's  weight  are  technicalities  of  trade  and  are 
not  needed  in  common  life. 

The  child's  original  motive  for  studying  these  facts  should  be 
found  largely  in  the  needs  arising  in  manual  training  and  other 
studies,  and  in  home  experience;  but  the  teacher  should  now  col- 
lect the  various  facts  into  "  tables,"  and  drill  upon  them  until  they 
are  properly  fixed. 

(d)  Compound  numbers  are  involved  in  the  measures  men- 
tioned.   In  this  grade  the  operations  with  them  are  confined  chiefly 
to  such  simple  reductions  as  are  necessary  in  the  applied  problems 
mentioned ;  namely,  feet  to  inches,  pounds  to  ounces,  days  to  weeks, 
and  similar  cases. 

(e)  Operations  with  integers.     The  essentially  new  feature 
of  the  year  is  the  completion  of  "  long  division."    The  early  printed 
arithmetics  gave  several  methods  of  division,  and  the  one  now 
commonly  used  was  finally  adopted  nearly  two  centuries  after  the 
beginning  of  printing,  showing  that  it  was  by  no  means  universally 
accepted  as  the  best.     Now  a  slight  improvement  on  the  common 
method  has  been  suggested,  and  is  rapidly  meeting  with  favor. 
It  is  one  of  the  forms  often  called  by  the  name  of  "  Austrian 
Method."     The  successive  steps  for  explaining  "  long  division  " 
vary  with  the  class.    In  a  general  way,  they  cover  the  following 
ground : 

Write  Think 

78 


3)234 

210 


24  3   210  +  24 


24  70+8 


The  quotient  of  200  -r-  3  is  not  any  number  of  loo's.  The 
quotient  of  23  tens  -r-  3  is  7  tens.  And  since  the  product  of  3  and 
7  tens  is  21  tens,  there  remains  24  to  be  divided.  24  -=-  3  =  8. 
And  since  the  product  of  3  and  8  is  24,  there  remains  nothing  to 
be  divided. 


129] 


Outline  for  the  First  Five  Grades 


39 


Similarly  for  625  -=-  25  : 
Write 

25 

25)625 
500 

125 

125 


Think 


25  [500+  125 
20+      5 


After  this  is  understood,  the  abridgment  as  shown  in  the 
common  "  Austrian  Method  "  should  be  used.     For  example :  — 

25 


25)625 
50 

125 
125 

While  the  decimal  fraction  is  not  taken  up  in  this  grade,  an 
example  involving  such  fractions  is  given  to  show  more  fully 
the  reason  for  adopting  this  particular  algorism. 

Required  to  divide  6.275  by  2.5 : 


OLD  METHOD 
2.5)6.275(2.51 
50 


I   27 
I   25 


COMMON  AUSTRIAN  METHOD 

2.51 
25)62.75 

50 


25 
25 


"  Point  off  as 
many  places  in  the 
quotient  as  the  num- 
ber of  decimal  places 
in  the  dividend  ex- 
ceeds that  in  the  di- 
visor." 


127 
125 

25 
25 

Dividend  and  divisor  having 
been  multiplied  by  such  a  power 
of  i  o  as  makes  the  divisor  a  whole 
number,  the  decimal  point  in  the 
quotient  simply  goes  above  that  in 
the  dividend. 


4O  Teachers  College  Record  [130 

The  following  method  is  recommended  for  the  early  work: 

2-51 
25)62.75 

50 


12-75 
12.5 

0.25 
0.25 

The  entire  remainder  is  brought  down  each  time,  and  the 
decimal  point  is  preserved  throughout. 

In  this  grade  the  child  is  ready  to  consider  the  twofold  notion 
of  division,  not  very  philosophically,  but  sufficiently  to  insure 
accuracy  of  statement.  Division  is  the  inverse  of  a  simpler  pro- 
cess, multiplication,  just  as  subtraction  is  the  inverse  of  a  simpler 
process,  addition.  A  complication  arises,  however,  in  division, 
which  does  not  arise  in  subtraction,  as  is  seen  in  the  following :  — 

Addition  and  subtraction :  — 

$3  +  $2  =  $5 ;  therefore,  (a)  $5  —  $3  =  $2 
or,  (b)  $5  — $2  =  $3. 

These  two  inverses  are  essentially  the  same,  because  the  num- 
bers all  represent  dollars. 

Multiplication  and  division :  — 

2  X  $3  =  $6 ;  therefore,  (c)  $6  -^  $3  =  2 
or,  (d)  $6  -H2  =  $3. 

These  two  inverses  are  different,  because  the  numbers  do  not 
all  represent  dollars,  as  in  the  case  of  addition  and  subtraction. 

These  two  cases  have  been  dignified  by  different  names,  the 
first  (c)  being  called  "  measuring "  and  the  second  (d)  "  par- 
tition." But  it  is  unwise  to  use  these  names  with  children.  The 
two  cases  are  easily  understood  by  children  sufficiently  to  lead 
them  to  the  use  of  correct  forms,  if  only  the  teacher  will  use  them. 

(f)  Fractions.  The  year's  work  includes  the  addition  and 
subtraction  of  common  fractions,  generally  with  one  figure  in 
each  term.  Simple  cases  of  multiplying  fractions  by  integers  are 
introduced.  In  many  schools  children  in  Grade  IV  go  beyond 
this  limit,  but  the  eventual  gain  by  so  doing  is  not  sufficient  to 
overcome  the  loss  in  interest  by  carrying  the  child  beyond  his 


131]  Outline  for  the  First  Five  Grades  41 

ordinary  needs  in  this  year.  Improper  fractions  and  mixed  num- 
bers are  also  introduced  in  this  grade. 

Sufficient  work  in  factoring  is  introduced  to  allow  the 
reduction  of  relatively  simple  fractions  to  lowest  terms.  This 
includes  the  tests  of  divisibility  by  2,  3,  5,  inductively  presented. 
The  idea  of  cancellation  is  also  introduced. 

The  question  as  to  the  precedence  of  decimal  and  common 
fractions  demands  consideration  at  this  point,  although  the  former 
is  met  only  incidentally  in  this  grade.  It  is  often  argued  that 
the  decimal  fraction,  being  merely  an  elaboration  of  the  deci- 
mal notation  with  integers,  should  be  taken  up  along  with  the 
latter;  that  since  United  States  money  is  taken  up  in  Grades  I 
and  II,  decimal  fractions  also  belong  in  those  grades.  But  this 
is  on  the  old  theory  that  when  a  topic  is  first  met  it  must  then 
be  mastered.  It  is  one  thing  to  know  how  to  indicate  values  in 
United  States  money,  and  quite  another  thing  to  know  how  to 
divide  one  decimal  fraction  by  another. 

The  needs  of  the  child  in  Grade  III  require  that  he  be  shown 
how  to  write  $1.25,  and  what  the  symbol  means.  They  do  not 
require  that  he  know  how  to  operate  with  decimal  fractions, 
although  he  should  know  that  y2  of  */2  is  %  ;  y2  of  %  is  l/^.  The 
idea  of  the  decimal  fraction  is  relatively  so  abstract  that  is  was  not 
until  well  along  in  the  i8th  century,  fully  one  hundred  fifty  years 
after  the  first  general  use  of  the  decimal  point,  that  this  form  of 
the  fraction  found  much  favor  in  the  business  world  or  in  the 
schools.  Hence,  it  cannot  be  expected  that  children  will  readily 
grasp  the  idea  of  decimal  fractions  before  the  idea  of  common 
fractions  is  fairly  well  fixed  in  mind.  Both  present  needs  and 
propaedeutic  considerations  demand,  however,  that  there  be  con- 
tinued practice  in  the  writing  of  numbers  in  dollars  and  cents, 
and  in  the  incidental  use  of  such  forms  as  0.50  and  $0%,  as  set 
forth  under  (&)  "Tables." 

(g)  Business  forms.  The  simple  form  of  bill,  as  used  in 
retail  trade,  is  introduced  in  this  grade,  for  the  reason  that  children 
at  about  this  period  begin  practically  to  meet  it  in  their  reading, 
and  in  purchasing  articles  at  stores. 

(h)  Oral  Drill.  There  should  be  in  this,  as  in  every  other 
grade,  frequent  drill  in  the  addition  of  columns  of  figures,  with 
a  view  to  accuracy  and  fair  rapidity;  and  in  subtraction  by  the 
"  making  change  "  method.  The  three  parts  of  arithmetic  that 


42  Teachers  College  Record  [132 

people  most  often  use  are  (i)  addition  of  columns  of  figures, 
(2)  making  change,  (3)  the  multiplication  table;  and  a  school 
day  should  seldom  pass  without  some  brisk  exercise  in  these 
subjects.  Above  all,  children  should  be  urged  to  read  columns 
of  figures  as  they  read  words  and  sentences;  they  do  not  spell 
the  words,  neither  should  they  slowly  count  up  a  column. 

(*)  Particular  attention  is  given  to  the  simple  problems 
involving  the  use  of  United  States  money,  manual  training,  the 
measurement  of  rooms  for  papering  and  carpeting  (but  without 
complications  involving  fractional  widths  and  patterns,  which  prac- 
tically are  left  to  dealers),  the  common  daily  purchases  of  a 
family,  and  the  common  occupations.  Maps  are  drawn  to  a  scale. 
Much  attention  is  given  to  accuracy,  to  a  fair  degree  of  speed, 
and  to  neatness  of  work. 

Grade  V 

i.  General  Suggestions 

(a)  Review  the  work  of  Grade  IV,  as  suggested  in  the  notes 
for  that  grade. 

(b)  On  the  use  of  the  text-book,  see  General  Suggestion 
(b)  under  Grade  III. 

(c)  The  work  of  the  year  is  chiefly  directed  to  the  funda- 
mental operations  with  compound  numbers  and  with  fractions, 
and  to  the  natural  extension  of  the  applications  of  arithmetic  to 
various  occupations,  and  to  other  quantitative  phases  of  life. 

(d)  For  further  suggestions  see  Grades  I-IV. 

(e)  Bibliography  :  —  Smith's  Teaching  of  Elementary  Math- 
ematics, as  noted  in  preceding  grades.    On  the  geometry :  Sundara 
Row's  Geometric  Paper  Folding,  for  the  spirit  of  the  work ;  also, 
Beman  and  Smith:  Higher  Arithmetic,  p.  66.     On  the  formal 
statement  of  the  work:  Beman  and  Smith:  Higher  Arithmetic, 
p.  41. 

2.  The  Mathematical  Work 

(a)  Compound  numbers.  All  of  the  necessary  tables  have 
now  been  taken.  The  reasons  for  omitting  certain  of  those  that 
formerly  had  place,  have  been  stated.  The  kilowat  and  horse- 
power, as  units  of  measure,  are  spoken  of  by  more  people  in 
New  York  City  to-day  than  are  the  scruple  and  the  link ;  so  that  if 


133]  Outline  for  the  First  Five  Grades  43 

we  were  to  add  to  the  number  of  the  tables,  it  would  not  be  in 
the  direction  of  the  apothecary's  weight  or  the  surveyor's  measure, 
but  rather  in  the  direction  of  the  newer  units  and  the  metric 
system.  These,  however,  if  taken  at  all,  should  enter  into  the 
work  of  the  later  grades,  where  the  pupil's  interest  demands 
them. 

The  reduction  of  compound  numbers  to  units  of  higher  and 
lower  denominations,  "  ascending "  and  "  descending,"  is  con- 
fined to  numbers  of  not  more  than  three  denominations.  The 
reasons  for  this  are  that  in  practical  life  we  rarely  use  more  than 
two,  as  feet  and  inches,  yards  and  inches,  pounds  and  ounces ; 
and  that  one  who  can  perform  reductions  with  three  denomina- 
tions can  easily  perform  those  with  more,  if  occasion  ever  demands. 
The  recent  change  in  custom,  with  respect  to  compound  numbers, 
is  quite  marked.  But  a  relatively  short  time  ago  it  was  not 
uncommon  to  see  the  area  of  a  field  stated  in  acres  and  rods, 
while  now  it  is  in  acres  and  decimal  parts  of  an  acre ;  lengths  were 
stated  in  rods  and  feet,  but  now  in  feet  and  decimals;  and,  in 
general,  compound  numbers  were  far  more  extensively  used  a 
generation  ago  than  now. 

The  operations  with  compound  numbers,  generally  involving 
only  two  denominations,  and  limited  to  three,  are  taken  in  this 
grade  as  suggested.  The  reasons  for  this  limitation  have  just  been 
given.  When  one  considers  the  rarity  of  occasions  for  the  use 
of  such  numbers,  by  himself  or  in  ordinary  business,  he  will  be 
convinced  that  the  time  formerly  devoted  to  the  subject  might 
better  be  spent  on  other  portions  of  arithmetic,  or  on  other 
subjects. 

(b)  Reduction  of  common  fractions.  The  growth  in  the  use 
of  the  decimal  fraction  has  been  so  great  during  the  past  century, 
that  much  of  the  work  formerly  necessary  in  common  fractions 
has  now  become  almost  obsolete.  Text-books  have  been  rather 
slow  in  recognizing  these  changes,  usually  being  followers  rather 
than  leaders  in  any  movement  of  this  nature.  It  therefore  becomes 
necessary  for  the  teacher  to  orientate  himself  somewhat,  before 
undertaking  the  work  in  fractions  per  se. 

When  arithmetics  began  to  be  printed,  the  sexagesimal  frac- 
tion was  used  for  all  scientific  purposes,  as  we  use  it  now  in 
degrees,  minutes,  and  seconds.  The  common  needs  of  trade 
demanded  another  fraction,  the  form  of  which  was  brought  from 


44  Teachers  College  Record  [134 

the  Arabs,  and  this  was  known  as  the  common  fraction.  In 
cases  where  we,  for  example,  now  would  use  fractions  like  0.432 
and  0.9375,  they  were  compelled  to  use  jYs  and  ||  .  If  these  frac- 
tions should  appear  as  |^f  and  £$£  it  was  desirable  to  reduce  them 
to  lowest  terms,  and,  if  they  were  to  be  added,  it  was  necessary 
to  reduce  them  to  common  denominators,  and  preferably  to  the 
least  common  denominators.  Hence  arose  the  necessity  for  a 
study  of  fractions,  and  for  finding  the  greatest  common  divisor 
and  the  least  common  multiple  of  two  or  more  numbers.  Now 
that  the  common  fraction  is  used  in  only  a  few  denominations, 
and  those  very  small,  the  decimal  fraction  becoming  more  com- 
mon in  general,  and  nearly  universal  in  scientific  and  monetary 
computations,  there  is  no  longer  the  practical  necessity  for  any 
extensive  treatment  of  factoring,  and  of  divisors  and  multiples, 
on  the  part  of  children. 

Hence  the  subject  of  factoring  is  limited  to  the  ability  to 
detect  the  factors  2,  3,  5,  these  being  sufficient  for  any  ordinary 
reduction  of  fractions  to  lowest  terms.  The  subject  of  greatest 
common  divisor  is  omitted,  as  a  topic,  for  the  reason  that  frac- 
tions formerly  requiring  the  use  of  such  divisors,  are  now  reduced 
to  the  decimal  form  before  operating,  and  the  "  practical  (applied) 
problems  "  are  always  so  palpably  artificial  as  to  be  unworthy 
of  attention.  The  subject  of  greatest  common  divisor  should 
receive  some  attention,  however,  for  the  reason  that  the  general 
information  of  pupils  may  still  demand  it  for  a  time,  but  it  is 
limited  to  the  treatment  by  factoring  as  stated.  There  is  a  little 
more  demand  for  the  least  common  multiple,  since  this  is  necessary 
for  finding  the  least  common  denominator  of  several  fractions, 
and  the  addition  and  subtraction  of  common  fractions  is  still 
important,  within  certain  limits.  For  this,  however,  the  factoring 
method  is  again  ample. 

(c)  Operations  with  common  fractions.  While  the  conven- 
tional order  of  treatment  is  possibly  justified  on  the  ground  of 
difficulty,  and  should  probably  be  followed ;  it  should  not  be  felt 
to  be  the  order  of  importance,  and  it  is  a  fair  question  as  to 
whether  it  is  the  order  of  difficulty.  Other  things  being  equal, 
addition  and  subtraction  of  common  fractions  are  simpler  than 
multiplication,  when  the  denominators  are  alike,  but  otherwise 
they  are  usually  not  so.  The  operation  2/3  of  6/7  is  easier  to 
perform  than  that  of  5/7  —  2/3,  and  it  is  more  important;  while 


135]  Outline  for  the  First  Five  Grades  45 

the  explanation  of  one  is  about  as  difficult  as  that  of  the  other. 
The  natural  order  of  presentation,  however,  would  seem  to  be: 
Addition  and  subtraction  of  fractions  having  the  same  denomi- 
nator; addition  and  subtraction  of  fractions  not  having  the  same 
denominator,  and  hence  the  necessity  for  a  common  denominator, 
and  in  particular  for  the  least  common  denominator;  multipli- 
cation; division. 

It  should  be  remembered,  however,  that  the  most  important 
of  these  operations  is  that  of  multiplication,  the  others  being 
used  comparatively  seldom.  That  is,  we  use  "  4  times  i2l/2  cents," 
and  expressions  similar  to  this,  more  often  than  we  add  common 
fractions.  It  should  also  be  remembered  that  however  much  we 
may  be  tempted  to  use  other  devices  for  division,  like  that  of 
reducing  to  a  common  denominator  and  then  dividing  numerators, 
the  practical  plan  is  that  of  multiplying  by  the  inverted  divisor. 
This,  therefore,  should  be  the  one  with  which  children  should 
become  familiar;  sufficient  explanation  being  given  to  justify  the 
process  to  their  minds,  but  no  elaborate  explanation  being 
required. 

(d)  Decimal  fractions.  As  already  stated,  these  are  carried 
along  with  common  fractions,  so  far  as  notation  is  concerned. 
But  it  now  becomes  necessary  to  study  them  per  se,  with  respect 
chiefly  to  the  operations. 

Having  become  more  or  less  familiar  with  the  notation, 
it  is  not  difficult  for  children  to  see  that  1/10,  2/20,  3/30,  10/ioo> 
o.io,  10%,  ten  hundredths,  ten  per  cent,  all  have  identically  the 
same  value,  and  all  mean  from  this  standpoint,  exactly  the  same 
thing.  Sometimes  it  is  more  convenient  to  write  1/10,  sometimes 
10%,  sometimes  o.io,  as  in  the  case  of  $2.10. 

The  addition  and  subtraction  of  decimal  fractions  usually 
offer  no  difficulties  unless  the  teacher  suggests  them  by  some 
attempt  at  over-explanation. 

The  difficulty  in  multiplication  has  often  been  made  the 
greater  by  the  attempt  to  refer  too  much  to  the  common- fraction 
notation.  The  most  approved  forms  are  the  following : 

(i)  The  decimal  points  are,  as  is  natural,  arranged  in  a 
column.  Since  5  times  hundredths  is  hundredths,  the  right  hand 
number  of  the  product  is  placed  under  hundredths.  The  rest  of 
the  work  is  identical  with  that  of  integers,  the  decimal  point 
going  under  the  others. 


46  Teachers  College  Record  [136 

6.25 

5- 


31-25 

(2)  In  this  case,  since  hundredths  multiplied  by  tenths  is 
thousandths,  the  right-hand  figure  of  the  product  goes  in  the 
thousandths  place. 

6.25 
0.5 


3-125 

(3)  In  this  case,  since  hundredths  multiplied  by  hundredths 
is  ten-thousandths,  the  right-hand  figure  of  the  product  goes 
in  the  ten-thousandths  place. 

6.25 
0.25 


0.3125 
1.250 

1-5625 

Not  only  is  this  arrangement  the  simplest  and  most  natural, 
but  there  is  an  ulterior  value  that  will  be  appreciated  by  computers. 
Since  a  large  amount  of  computation  in  scientific  matters  need 
be  carried  to  only  two  or  three  decimal  places,  this  arrangement 
is  the  best  for  the  approximate  multiplications  demanded.  Thus 
in  the  approximate  multiplication  of  10.48  by  3.1416  we  have, 
multiplying  first  by  the  units: 

10.48 
3.1416 


31-44 
1.048 
0.419 

O.OIO 

0.006 

32-92 

The  result   is   correct  to  two  places,   and   the   unnecessary 
work  is  eliminated.     The  saving  is  more  pronounced  in  more 


137]  Outline  for  the  First  Five  Grades  47 

elaborate  problems.     Of  course  this  latter  work  is  not  intended 
for  this  grade. 

The  division  of  decimal  fractions  has  already  been  men- 
tioned in  Grade  IV,  2,  (e). 

The  operations  with  decimals  are  generally  limited,  for  rea- 
sons already  stated,  to  fractions  having  not  more  than  three  places. 

Such  simple  aliquot  parts  (the  name  is  not  used)  as  .33^, 
.66%,  .20,  .25,  are  taken  up  in  this  grade. 

(e)  Business  forms.  Carrying  forward  the  work  of  Grade 
IV,  attention  is  given  to  the  bill,  the  receipt,  and  the  meaning  of 
debit  and  credit. 

(/)  Oral  work.     See  Grade  IV,  2,  (h). 

(g)  Geometry.  The  subject  of  geometry  in  the  grades 
has  of  late  attracted  much  attention.  The  nature  of  the  work 
depends  upon  the  answer  to  the  question,  Why  should  it  be  there  ? 
Its  presence  might  be  justified,  (i)  because  the  child's  interests" 
demand  it,  (2)  because  the  child  needs  the  logical  training  which 
it  offers,  (3)  because  general  information  demands  it,  or  for 
similar  reasons,  all  more  or  less  connected,  and  all  summed  up 
in  the  statement  that  the  child  is  ready  for  it.  But  for  what  is 
the  child  ready?  For  a  knowledge  of  geometric  forms  certainly, 
and  this  he  has  with  his  manual  training  all  through  the  grades, 
learning  the  names  of  simple  geometric  figures  as  a  part  of 
that  work.  It  may  be  doubted  if  he  needs  to  know  of  any  other 
forms  of  plane  geometry  than  those  met  in  that  way. 

Furthermore  he  has  interest  in  knowing  how  to  measure 
certain  of  these  forms,  such  as  the  parallelepiped  and  prism,  which 
may  be  met  even  in  sewing.  In  other  words,  mensuration  should 
be  undertaken  rationally,  such  concrete  problems  being  solved  as 
are  actually  met  in  working  with  material  things,  this  being  the 
"  laboratory  method  "  in  its  truest  form. 

But  he  does  not  need  at  present,  and  he  is  not  yet  prepared 
for,  the  deductive  proof  of  geometric  propositions.  Proof  by 
things.  As  before  stated,  however,  the  subject  of  geometry  is 
actual  trial,  and  proof  by  logical  reasoning,  are  two  very  different 
reserved  for  a  later  article. 

So  much  being  premised,  the  work  in  this  grade  is  limited 
to  the  proper  naming  of  the  common  geometric  forms,  particu- 
larly as  met  in  the  manual  training,  and  to  the  mensuration  of 
certain  of  these  forms,  as  follows : 


48 


Length :  circumference  of  circle  compared  with  the  diameter. 

Area:  square,  rectangle,  parallelogram,  triangle,  circle. 

Volume:  rectangular  parallelepiped  and  prism. 

Much  interest  may,  however,  be  awakened  by  extending 
this  work  to  the  construction  of  regular  polygons;  tiles  or  hard- 
wood floors  offering  an  example  of  the  hexagon,  and  the  bee's 
cell  showing  its  economic  value. 

It  is  very  desirable  to  hold  to  the  unity  of  mathematics,  not 
letting  the  child  feel  that  his  study  of  geometry  is  separate  from 
the  study  of  arithmetic.  Both  are  mathematics  and  are  inter- 
related in  life;  the  school  too  often  separates  such  close  inter- 
relations. 

(h)  Formal  work.  Children  now  approach  the  stage  of 
more  formal  reasoning  about  the  work.  As  a  consequence,  the 
solution  of  applied  problems  should  now  appear  in  steps,  neatly 
arranged  and  numbered.  Accuracy  of  form  is  necessary  to 
accuracy  of  reasoning.  The  following  will  illustrate  what  is 
meant : 


COMMON  INACCURACIES 

1.  2  X  25  —  $50. 

2.  2  X  $25  =  $50.  +  $2.  =  $52. 

3.  2  ft.  X  4  ft.  =  8  ft.,  or  8  sq.  ft. 

4.  100  -4-  4  =  $25. 

5.  $100-1-4  =  25. 

6.  100  -H  $4  =  $25. 

7.  V  144  sq.  ft.  —  12  ft. 

8.  "  As  many  times  as  4  is  con- 

tained in  $100." 

9.  "  Tens  o'  thousandths." 


THE  ACCURATE  FORM 

1.  2  X  $25  —  $50. 

2.  2  X  $25  =  $50 

$50  4-  $2  =  $52. 

3.  2  X  4  sq.  ft.  =  8  sq.  ft.,  or 

2  X  4  X  I  sq.  ft.  =  8  sq.  ft. 

4.  $100  —  4  =  $25. 

5.  $100  —  $4  =  25,  or 
$100  —  4  =  $25. 

6.  Impossible.    See  5. 

7.  V*44  ft.  =  12  ft.,  or 

VH4X  i  ft.  =  12  ft. 

8.  "  As  many  times  as  $4  is  con- 

tained in  $100  "  or  "  as  4 
is  contained  in  $100." 

9.  "  Ten  thousandths." 


III.  GENERAL  DISCUSSION  OF  THE  WORK  OF  THE 
LAST  THREE  GRADES 

(a)   MODERN  SUBJECT-MATTER 

The  work  of  the  last  three  grades  is  chiefly  devoted  to  the 
application  of  arithmetic  to  the  affairs  of  life.  It  is,  therefore, 
desirable  to  know  the  nature  of  the  problems  demanded  by  the 
business  of  to-day,  by  the  science,  manual  training,  and  other 
subjects  in  related  courses  in  the  school.  This  matter  needs  there- 
fore to  be  considered  at  some  length  from  the  point  of  view  of  the 
teacher. 

Knowing  thus,  in  general,  the  kind  of  subjects  that  should 
constitute  the  principal  topics  for  study  in  the  sixth,  seventh  and 
eighth  grades  of  school,  we  have  a  standard  for  judging  the 
worth  of  the  traditional  topics  and  processes  customarily  dealt 
with  in  these  years.  In  other  words,  we  can  decide  as  to  what  is 
really  important  in  social  life,  and  what  is  really  used;  and  we 
can  also  decide  what  is  not  so  useful,  and  what  may  even  be  cast 
aside  as  having  slight  value. 

Thus  far  in  the  grades,  the  mathematics  has  had  two  promi- 
nent purposes,  so  far  as  knowledge  is  concerned;  (a)  acquaint- 
ance with  certain  processes;  and  (b)  acquaintance  with  valuable 
facts  of  a  quantitative  nature.  To  the  teacher  the  former  may 
rightly  have  been  a  prominent  object;  just  as  in  teaching  litera- 
ture the  inculcation  of  the  underlying  truth  may  be  the  great 
purpose.  But  to  the  child,  the  answers  to  problems,  if  the  problems 
have  been  well  chosen,  may  have  been  the  attractive  aim ;  just  as 
in  literature  it  is  the  incidents  with  their  "  outcome,"  in  the  narra- 
tive that  he  reads.  In  other  words,  the  teacher's  object  may  be 
quite  different  from  that  of  the  child;  and  very  often  it  is  so. 

From  the  beginning  of  the  sixth  grade,  the  mathematical 

processes  as  objects  of  thought  should  grow  less  prominent  in 

the  minds  of  both  teacher  and  pupil,  for  the  reason  that  most  of 

these  processes  have  already  been  mastered ;  the  other  phase  of 

139]  49 


50  Teachers  College  Record  [140 

the  work,  meanwhile,  should  increase  in  prominence.  In  other 
words,  the  primary  aim  of  teacher  and  pupil  should  now  be 
increasingly  a  study  of  social  and  business  life.  Just  as  literature 
and  history  teach  ethical  and  political  principles  of  conduct;  so 
arithmetic  should  now  teach  quantitative  principles  of  business, 
and  problems  are  the  means  for  arriving  at  this  knowledge. 

In  outlining  the  work  for  the  next  three  grades,  therefore, 
it  is  our  duty  to  name  not  primarily  the  arithmetical  processes 
to  be  treated,  but  the  large  topics  that  must  be  investigated  on 
the  quantitative  side.  Some  of  these,  for  example,  are  transporta- 
tion by  rail,  by  canal  and  ocean,  with  comparisons ;  corporations ; 
government  revenues  and  expenditures;  manufacturing;  farming. 
Modern  arithmetics  already  show  a  tendency  in  this  direction; 
for,  while  they  discuss  topics  like  ratio  and  proportion,  discount, 
interest,  profit  and  loss,  which  are  mainly  concerned  with  proc- 
esses, they  also  take  up  topics  like  banking,  stocks  and  bonds, 
and  commissions,  which  are  definite  kinds  of  business.  In  fact, 
a  modern  arithmetic  is  a  confused  mixture  of  processes,  and  of 
the  occupations  of  men,  although  it  is  evident  all  the  time  that 
there  are  very  few  new  processes  that  need  be  taught.  Our 
proposal  amounts  only  to  the  suggestion  that  this  confusion  be 
remedied  by  a  fuller  acceptance  of  great  business  enterprises  as 
topics  of  study. 

And  if  this  were  done,  the  leading  defect  of  our  present 
study  of  arithmetic  would  be  largely  overcome.  For  every  one 
now  has  the  feeling  that  children  begin  to  figure  on  a  given 
problem  too  quickly,  although  it  is  certain  that  the  work  is 
none  too  accurate;  they  are  at  work  with  the  pencil  with 
altogether  too  little  thought  of  the  general  conditions  under 
which  the  actual  business  is  conducted.  This  is  because  our 
very  definition  of  arithmetic  has  been  "  figuring."  If  we 
adopted  the  other  conception,  namely,  that  arithmetic  is  a  study 
aiming  to  bring  one  into  an  understanding  and  appreciation  of 
leading  business  undertakings,  the  emphasis  would  naturally  fall 
on  the  nature  of  each  undertaking.  This  would  insure  more 
intelligence  from  the  start,  and  give  motive  for  the  mechan- 
ical work  in  the  solution  of  problems.  The  effect,  too,  would 
be  that  the  processes  themselves  would  be  better  understood ;  just 
as  the  conventional  forms  in  written  language  are  better  learned 
when  children  have  really  interesting  subjects  to  write  about. 


141]  Work  of  the  Last  Three  Grades  51 

Aside  from  this  argument,  while  many  of  these  topics  are 
among  the  most  important  themes  of  daily  conversation,  they  are 
not  and  cannot  be  provided  for  in  other  studies,  because  an  under- 
standing of  them  is  reached  mainly  by  a  study  of  quantity.  It 
seems  strange,  under  such  circumstances,  that  arithmetic  has  been 
so  long  willing  to  confine  itself  exclusively  to  processes.  With 
the  examples  of  reform  set  by  other  studies  along  the  same  line, 
notably  language  work  in  its  endeavor  to  enliven  form  through 
a  richer  content,  it  is  fitting  that  a  forward  movement  be  made 
in  arithmetic.  Interest,  therefore,  may  be  as  great  an  aim  in 
arithmetic  as  in  literature  or  in  nature  study;  for  the  field  of 
thought  may  be  as  inspiring. 

Hence  it  is  well  to  begin  by  enumerating  some  of  the  principal 
business  undertakings  that  should  constitute  the  course  of  study 
in  arithmetic  in  the  sixth,  seventh,  and  eighth  grades.  A  discus- 
sion of  the  various  obsolete  and  of  the  acceptable  methods  or 
processes  will  next  be  in  order.  Then  the  special  work  of  each 
year  will  follow. 

Topics  for  study  from  which  selections  might  well  be  made 
for  the  sixth,  seventh,  and  eighth  grades  are  as  follows :  — 

Farm  life:  Dairy  farm  in  New  England;  dimensions  of  a 
typical  farm  ;  cattle ;  milk ;  prices ;  butter ;  feed.  Size  of  typical 
farm  in  Illinois ;  typical  space  given  to  each  product ;  quantity  of 
this  product.  Irrigated  farms  in  Colorado;  size;  quantity  and 
price  of  water;  variety  and  quantity  of  products  (compare  with 
the  two  former)  ;  effect  of  irrigation  on  the  price  of  land.  Wheat 
farm  in  Washington ;  area ;  quantity ;  cost  of  production ;  prices ; 
manner  of  renting  land  ;  variation  from  year  to  year.  Fruit  farms 
in  New  York ;  in  Southern  California ;  in  New  Jersey ;  in  Florida. 
Cotton,  rice,  and  sugar  plantations. 

Rainfall :  Variations  in  different  places  and  from  year  to 
year. 

Mining:  Nature  of  a  mine;  coal  mine;  dimensions;  number 
of  men ;  amount  of  coal ;  bituminous  and  anthracite  mines  com- 
pared. Iron  ore  in  our  own  states ;  comparison  with  Great  Britain. 
Copper  output  of  Calumet  and  Hecla  Mines ;  general  comparisons. 
Gold  and  silver  mining. 

The  Oil  Industry :  variety  and  quantity  of  products ;  com- 
parison with  Russia. 

Lumbering. 


52  Teachers  College  Record  [142 

Fishing. 

Ranching:  Cattle  ranch;  sheep  ranch;  dimensions;  losses; 
prices. 

Transportation :  Wagon  roads ;  extent  and  cost  of  various 
kinds  of  roads  and  pavement  in  various  states  and  cities.  Canal ; 
extent ;  history  of  the  Erie  Canal  on  the  quantitative  side ;  amount 
of  freight  carried.  Pennsylvania  Railroad  system ;  freight  on  the 
N.  Y.  Central  Railroad  compared  with  that  on  the  Erie  Canal; 
that  on  the  Illinois  Central  Railroad  compared  with  that  on  the 
Mississippi  River.  Ocean  traffic ;  size  of  steamships ;  freight 
capacity ;  coal  consumption.  Elevators  for  grain. 

Manufacture:  Cotton;  comparison  of  factories  in  the  South 
and  in  Massachusetts;  wages;  prices.  Wool;  herd  of  sheep  in 
Ohio;  in  Nebraska;  factories;  wages;  prices.  Leather.  Clothing 
in  New  York ;  the  sweat-shops.  Sugar ;  cane  and  beet  sugar 
compared.  The  Linen  industry.  The  Silk  industry.  Gas  com- 
pared with  electricity  as  to  cost;  price  of  each  in  different 
cities. 

Sale  of  Commodities:  Grocery  Store.  Bakery.  Department 
Store.  Wholesale  Store;  dimensions;  stock;  traveling  men. 

Banking  business :  deposits ;  loans ;  exchange. 

Corporations  :  stocks ;  bonds. 

Rents :  How  farms  are  rented ;  amounts.  Tenement  houses 
in  New  York,  with  reference  to  recent  reforms.  Apartments  in 
New  York.  Houses.  Office  buildings. 

Insurance:  fire;  accident;  life. 

Revenues  and  expenses  of  New  York  City :  taxes  of  various 
kinds;  departments;  employees;  supplies;  cost  of  elections;  com- 
parisons with  other  cities  at  home  and  abroad. 

Revenues  and  expenses  of  the  United  States  Government: 
Patent  Office  Department.  Treasury  Department;  internal 
revenue ;  tariff ;  imports  and  exports ;  custom  houses.  War 
Department;  its  size;  the  cost  of  maintaining.  Pension  Depart- 
ment. Navy  Department;  warships;  cost  of  navy.  The  capitol, 
and  the  various  other  expenses,  such  as  those  of  the  White  House 
and  the  Department  of  State. 

Immigration :  Nationalities  ;  restrictions. 

Power  for  manufacturing;  horse-power  required  in  various 
factories;  electricity  and  steam;  Niagara  Falls  as  a  generator  of 
power. 


143]  Work  of  the  Last  Three  Grades  53 

Labor :  The  cost  at  home  and  abroad ;  in  various  occupations. 

Education :  Cost  of  our  schools ;  a  great  library ;  a  university. 

The  cost  of  printing:  The  extent  of  the  industry;  consump- 
tion of  ink  and  paper. 

Public  health :  The  effect  of  improved  sewerage,  of  good 
water,  of  clean  streets,  of  better  ventilation. 

The  investigation  of  almost  any  one  of  these  topics  on  the 
quantitative  side  calls  into  use  many,  and  perhaps  all,  of  the 
arithmetical  processes  that  have  thus  far  become  familiar  to  the 
pupil.  In  arithmetic,  as  in  geography,  children  are  in  the  habit 
of  forgetting  one  subject  while  learning  another;  but  the  above 
arrangement  renders  such  forgetting  impossible  in  arithmetic. 
Aside  from  that,  however,  text-books  in  arithmetic  have  too  few 
"  promiscuous  examples."  The  examples  and  problems  ordinarily 
bear  so  exclusively  on  one  process  that  the  pupils  drop  into  a 
mechanical  application  of  a  single  new  rule  before  the  series  of 
examples  is  half  completed,  and  the  remainder  of  the  problems 
are  solved  with  the  minimum  amount  of  thought.  A  greater 
variety  of  work  would  prevent  the  application  of  the  new  rule, 
or  of  any  particular  new  thought,  to  every  example,  and  would 
thus  compel  the  pupil  to  think  at  every  step  of  his  progress. 

(&)  TRADITIONAL  SUBJECT-MATTER 

The  preceding  discussion  suggests  an  examination  into  the 
traditional  topics  of  the  more  advanced  arithmetics,  with  a  view 
to  determining  their  value.  This  examination  must,  in  this  case, 
be  very  brief  and  limited  to  the  more  prominent  topics. 

(i)  Percentage.  This  subject,  formerly  a  topic  by  itself,  is 
merely  one  phase  of  decimal  fractions,  and  should  be  so  treated. 
A  large  part  of  business  arithmetic  involves  the  finding  of  per 
cents,  so  that  the  method  is  continually  applied  after  it  is  once 
presented.  The  treatment  of  the  subject  by  "  cases,"  and  the 
learning  of  definitions  of  terms  like  "  amount,"  "  difference,"  or 
even  "  percentage,"  may  be  considered  obsolete.  There  is  need 
to  know  what  "  per  cent  "  means,  namely,  "  hundredths  "  ("  hun- 
dredth," or  "of  a  hundredth,"  as  in  6%,  i%,  l/2%),  and  there 
is  occasionally  some  value  in  using  the  term  "  base."  But  the 
two  leading  problems  of  the  subject  are  illustrated  by  two 
examples  not  requiring  any  elaborate  vocabulary,  namely: 


54  Teachers  College  Record  [144 

1.  6%  of  $250  is  how  much? 

2.  If  104%  of  x  =  $7.28,  what  does  x  equal? 
Practical  problems  in  percentage  rarely  require  any  other 

forms. 

(2)  Discounts,     (a)   Commercial:  Of  high  value,  since  it  is 
used  in  all  business,  from  extensive  wholesale  transactions  down 
to  so-called  bargain  sales. 

(b)  Bank:  Generally  used  by  those  having  occasion  to  bor- 
row money  at  banks.    Of  high  value.    See  (8)  below. 

(c)  So-called  "  true  "  discount :  The  very  name  condemns  it. 
It  gives  no  true  idea  of  business  and  should  be  omitted.    It  should, 
of  course,  be  recognized  that  this  form  of  discount  is  more  advan- 
tageous to  the  borrower  than  bank  discount,  and  on  this  account 
it  has  had  many  champions  on  theoretic  grounds.     For  us,  how- 
ever, it  is  merely  a  question,  in  the  grades,  as  to  which  is  practi- 
cally used. 

(3)  Interest,      (a)   Simple:   Necessary  in  all  business  life. 
But  in  teaching  this,  as  other  similar  subjects,  the  teacher  needs 
to   look   at   the   matter   in   perspective.     That   is,   one   class   of 
problems  is  very  important;  several  others  are  relatively  unim- 
portant, and  the  teacher  should  present  the  subject  with  this  in 
mind.     The  important  point  is  that  the  pupils  should  know  what 
simple  interest  is  and  how  it  is  computed,  with  tables  and  with- 
out.    It  is  unimportant,  from  the  business  standpoint,  that  the 
pupil  should  be  able  to  find  the  time,  given  the  principal,  rate,  and 
interest.     Hence,  while  this  latter  case  is  valuable  as  fixing  the 
former  in  mind,  it  deserves  little  emphasis. 

(b)  Compound:   It  is  valuable  for  a  child  to  know  what 
compound  interest  is,  and  that  it  is  allowed  by  savings  banks  in 
most  states,  and  that  the  rate  of  interest  is  relatively  low.     It  is 
well  that  there  should  be  enough  exercise  in  computing  com- 
pound interest,  with  and  without  tables,  to  fix  the  idea  in  mind. 
This  being  accomplished,  the  subject  has  little  practical  value. 
To  those  who,  either  here  or  in  other  subjects  of  arithmetic,  plead 
for  mental  gymnastics,  it  should  again  be  said  that  arithmetic 
offers  ample  facilities  for  this  valuable  work  without  giving  a 
false  idea  of  business  customs  and  values.     Hence  the  value  of 
this  subject  is  relatively  low. 

(c)  Annual:  This  is  rarely  used  and  hence  is  of  low  value. 
When  met,  however,  in  coupon  notes,  it  involves  no  new  principle. 


I45]  Work  of  the  Last  Three  Grades  55 

(d)  Partial  Payments :  This  was  formerly  of  high  value,  but 
the  increase  of  banking  facilities,  and  the  simplification  of  banking 
methods,  have  diminished  this  value.  In  rural  communities  it  still 
has  considerable  importance,  the  "  United  States  Rule  "  being  the 
only  one  generally  recognized  in  most  states.  In  order  to  secure 
the  value  of  the  subject,  problems  should  involve  common  amounts, 
as  a  principal  of  $100  with  payments  like  $10  and  $25,  rather  than 
amounts  like  $251.42  and  $19.79.  If  complicated  multiplications 
are  desired,  let  them  enter  where  they  belong,  in  problems  involv- 
ing scientific  measurements.  To  introduce  them  here  is  to  make 
the  problem  both  unreal  and  uninteresting.  Taught  as  suggested, 
the  subject  has  a  moderate  value. 

(4)  Insurance.    This  is  important  in  three  general  lines :  fire, 
accident,  and  life  insurance.    As  to  the  first,  it  is  important  that 
one  class  of  problems  should  be  emphasized,  the  usual  one-year  arid 
three-year  policies,  at  the  prevailing  rates  in  this  part  of  the 
country,  the  premium  being  required.     As  to  the  second,  it  is 
important  that  the  system  be  explained  and  a  few  simple  problems 
solved.    As  to  the  third,  life  insurance,  the  classes  of  policies  are 
now  so  numerous  that  only  two  or  three  standard  cases  need  be 
considered.     The  "  straight  life,"  and  the  endowment  policies  are 
the  most  frequently  found,  and  these  should  be  considered  in  the 
light  of  the  premiums  and  the  returns.     (See  Beman  &  Smith's 
Higher    Arithmetic,    p.    177.)      The    value    of    this    subject    is 
high. 

(5)  Stocks  and  bonds.     Since,  in  these  days,  most  business 
requiring  large  capital  is  conducted  by  corporations,  and  the  ques- 
tion of  the  rights  and  duties  of  capital  are  vital  in  economics,  it 
is  very  necessary  to  relate  arithmetic  to  this  phase  of  modern  life. 
The  following  matter  should  be,  with  due  regard  to  sequence, 
explained  to  the  class :  Corporations,  stock  companies  being  special 
forms ;  directors  and  officers  of  stock  companies ;  stock,  preferred 
and  common ;  dividends ;  certificate  of  stock ;  bonds,  coupon  and 
registered;  interest;  par,  above  and  below  par;  usual  par  value 
of  a  share  of  stock  and  of  a  bond ;  how  stocks  and  bonds  are  pur- 
chased ;  stock  exchange ;  broker's  commissions ;  newspaper  quota- 
tions used  in  class ;  distinction  between  legitimate  and  illegitimate 
dealings  in  stock;  the  percentage  against  the  gambler. 

The  problems  should  be  confined  to  the  usual  cases,  and 
may  best  be  assigned  with  reference  to  the  newspaper  quotations. 


56  Teachers  College  Record  [146 

The  following  errors  are  not  uncommon  in  text-books,  and  should 
be  avoided :  Buying  a  fractional  part  of  a  share ;  paying  a  com- 
mission different  from  y%  per  cent;  sending  a  certain  sum  to  a 
broker  to  be  entirely  invested,  instead  of  ordering  the  broker  to 
buy  a  certain  number  of  shares  and  then  paying  him;  quotations 
differing  from  the  prevailing  prices  of  standard  stocks. 

Taken  up  in  the  modern  spirit,  with  blank  forms  of  stock 
certificates  and  of  bonds  to  be  seen  by  the  class,  the  subject  has 
high  economic  value,  although  mathematically  it  offers  nothing 
new.  (For  suggestions,  see  Beman  &  Smith's  Higher  Arith- 
metic, p.  171.) 

(6)  Commission   and   brokerage.      In   this   is   involved   the 
element  of  the  question  of  food  supply  for  a  city  like  New  York, 
and  from  this  practical  standpoint  it  should  be  approached.    The 
practical  mathematical  side  of  the  case  is  simple,  and  the  non- 
practical  side,  involving  unusual  business  problems,  should  not 
be  considered.    (See  Beman  &  Smith's  Higher  Arithmetic,  p.  168.) 
The  economic  value  is  therefore  high,  although  no  new  mathe- 
matical processes  are  involved. 

(7)  Government  revenues.     Instead  of  isolating  the  chapter 
on  taxes,  it  is  much  better  to  consider  this  detail  in  connection 
with   government   revenues   in  general.     There   should   then  be 
explained  to  the  class  the  following:  The  expenses  of  the  United 
States  Government,  and  where  the  money  goes;  the  same  for  the 
state  and  the  city;  the  sources  of  income  to  meet  the  expenses  of 
the  United  States  Government ;  internal  revenues,  including  post- 
age, and  customs  duties ;  the  revenues  for  the  state  and  city ;  taxes 
on  real  and  personal  property ;  inheritance  taxes ;  prevailing  tax 
rates ;  construction  of  a  tax-table,  and  explanation  of  its  use. 

The  problems  should  then  relate  to  the  common  cases  of 
direct  taxation,  of  internal  revenue,  and  of  customs.  To  manu- 
facture unreal  problems  of  a  complicated  nature  is  to  put  out  of 
relief  the  practical  problems.  Mathematical  difficulties  are  better 
presented  in  other  ways.  From  the  standpoint  suggested  the 
subject  has  high  economic  and  rather  high  mathematical  value. 
(Beman  &  Smith's  Higher  Arithmetic,  p.  163.) 

(8)  Banking  business.    Arithmetics  quite  generally  separate 
bank   discount   from  exchange,  and   each  of  these  topics   from 
numerous  other  features  of  banking,  with  which  pupils  should 
become  acquainted.     The  ordinary  citizen  needs  to  know  some- 


147]  Work  of  the  Last  Three  Grades  57 

thing  of  the  following  subjects:  Savings  banks  versus  general 
commercial  banks ;  how  to  open  a  bank  account ;  deposit  slips  and 
checks;  certificates  of  deposit;  interest  on  deposits;  the  banking 
business  of  express  companies  and  the  post  office,  as  well  as  of 
banks,  as  shown  in  money  orders  and  drafts. 

If  one  is  in  business  he  will  need  to  know :  The  process  of 
borrowing  money  from  a  bank ;  bank  discount ;  security ;  discount- 
ing notes  that  he  may  hold ;  collecting  by  means  of  drafts ; 
sending  money  by  means  of  drafts. 

His  general  stock  of  information  should  also  include  a 
knowledge  of  the  clearing  houses  in  various  cities,  and  particu- 
larly in  New  York,  together  with  the  volume  of  business  trans- 
acted. 

From  this  standpoint  the  economic  and  mathematical  value 
of  the  subject  is  high.  (Beman  &  Smith's  Higher  Arithmetic, 

P-  148.) 

(9)  Profit  and  loss.    A  very  old  name  for  a  special  chapter. 
At  present  the  business  expression  "  Profit  and  loss "  has  an 
entirely  different  meaning  from  that  given  in  arithmetic.     Since 
the  chapter  involves  no  new  principles,  pertains  to  no  particular 
line  of  business,  and  involves  no  problems  not  naturally  to  be 
placed  in  miscellaneous  business  exercises,  it  may  well  be  omitted 
as  a  special  topic. 

(10)  Equation  of  payments.    An  old  chapter,  once  of  much 
practical  value,  but  nearly  obsolete  in  America.     Improved  bank- 
ing   facilities   have   made   these   long-standing   accounts    unnec- 
essary. 

(n)  Partnership.  This  was  formerly  a  very  valuable  sub- 
ject, particularly  before  the  invention  of  stock  companies.  Part- 
nerships still  exist;  they  are,  indeed,  more  numerous  than  ever, 
but  the  style  of  problem  set  forth  in  the  arithmetics,  under  this 
topic,  has  long  been  obsolete. 

(12)  Longitude  and  Time.  This  subject,  formerly  of  much 
practical  importance,  ceased  to  have  some  of  this  value  with  the 
general  adoption,  at  the  close  of  the  I9th  century,  of  standard 
time  in  a  large  part  of  the  civilized  world.  Practically,  it  is  at 
present  rather  a  part  of  geography  than  of  arithmetic.  Problems 
based  on  the  15°  scheme  should  predominate.  Other  problems 
should  generally  refer  to  ships  at  sea  and  to  observatories, 
becoming  thus  too  technical  for  the  grades. 


58  Teachers  College  Record  [148 

(c)  SUMMARY  CONCERNING  SUBJECT-MATTER 

These  conventional  subjects  have  been  mentioned  by  some- 
what the  same  names  found  in  the  ordinary  arithmetics,  so  that 
the  suggestions  may  be  of  service,  no  matter  what  book  may  be 
in  use.  The  change  in  the  standard  chapters  has  recently  been 
very  marked.  When  the  old  "  Rule  of  False,"  or  "  Rule  of  False 
Position,"  disappeared,  many  conservative  teachers  felt  that  there 
had  been  a  great  loss.  When  alligation  disappeared,  a  few  years 
ago,  there  was  a  similar  protest.  Now  that  equation  of  payments, 
and  compound  proportion,  and  unitary  analysis,  and  profit  and 
loss,  and  other  chapters,  are  going,  many  very  good  teachers  feel 
that  nothing  will  be  left  of  arithmetic.  But  in  place  of  every 
business  custom  or  mechanical  process  that  becomes  antiquated, 
new  customs  and  new  processes  appear,  and  for  these  we  need  to 
be  prepared.  As  the  opportunity  offers  the  following  newer  topics 
are  introduced,  the  problems  taken  from  current  literature: 

1.  The  question  of  the  agricultural  interests  of  the  country, 
connecting  with  commissions,  banking,  taxes,  and  transportation. 

2.  The  questions  of  fishing,  lumbering,  mining,   manufac- 
turing, and  trade. 

3.  The  question  of  transportation,  connecting  with  the  study 
of  corporations,  taxes,  agriculture,  mining,  and  banking. 

4.  The  questions  of  labor  and  of  labor  organizations,  and  the 
relation  of  each  to  the  questions  above  mentioned. 

5.  Modern  treatment  of  the  question  of  government  revenues 
and  expenditures. 

6.  Modern  treatment  of  the  problems  of  banking. 

7.  Modern  treatment  of  problems  involving  corporations,  as 
suggested  above  under  stocks  and  bonds. 

These  topics  are  alive  and  are  valuable  for  every  citizen.  It 
will  be  a  better  day  for  the  schools  when  they  replace  the  obsolete 
subjects  to  which  reference  has  been  made. 

It  is  important  to  note  that  the  topics  named  suggest  our 
broader  commercial,  industrial,  and  social  life  as  the  field  for 
arithmetic  in  the  higher  grades.  Thereby,  elementary  mathe- 
matics is  made  to  stand  on  the  same  plane  as  literature  and  other 
studies,  for  all  these  now  culminate  in  rich  generalizations.  We 
should  expect  in  each  grade,  therefore,  a  crude  formulation  of 
rules  of  business  rather  than  rules  for  arithmetical  processes. 


149]  Work  of  the  Last  Three  Grades  59 

For  example,  a  study  of  farming-  would  result  in  a  knowledge 
of  many  principles,  touching  quantity,  that  guide  the  farmer; 
such  as  the  proportionate  division  of  his  land  for  certain  crops, 
the  customary  amount  of  stock,  the  variation  in  prices  that  may 
be  expected.  A  study  of  rents  would  acquaint  the  pupil  with 
methods  of  renting  farms  and  tenements,  the  per  cent  of  profit 
than  can  be  expected,  the  dangers,  and  the  customary  losses. 
A  study  of  insurance  would  likewise  lead  to  some  knowledge  of 
the  way  in  which  risks  are  calculated,  what  rates  are  paid,  and 
the  provisions  a  man  should  make  for  the  support  of  his  family 
after  his  death.  A  text-book  should  word  these  principles  with 
the  same  care  with  which  rules  for  processes  have  heretofore 
been  worded.  Thus,  mathematics  in  the  grades  would  cease  to 
be  purely  theoretical;  but,  on  the  contrary,  would  lead  to  an 
understanding  of  practical  affairs. 

But  arithmetic  in  the  grades  can  easily  go  farther  than  this. 
Why  is  it  not  a  fair  task  to  require  a  seventh  or  eighth  grade 
pupil  to  discover  by  himself  how  much  it  costs  to  keep  a  horse 
or  cow  in  a  given  community,  or  to  estimate  how  much  butter 
a  cow  might  make,  averaging  2l/2  gallons  of  milk  per  day,  or  to 
find  the  cost  of  fuel  for  the  school,  or  to  investigate  other  similar 
matters?  Such  a  lesson  might  be  assigned  a  week  or  more  in 
advance,  with  the  understanding  that  each  pupil  should  collect 
the  most  reliable  data  he  could  get,  and  make  a  full  report, 
including  the  answer  to  the  problem,  to  the  class.  After  some 
experience,  the  children  might  be  left  to  their  own  resources  as 
to  where  to  go,  how  to  make  an  appointment  with  a  man,  what 
questions  to  ask,  how  to  know  when  all  necessary  questions  have 
been  put,  and  how  to  use  the  data  collected.  If  such  work  were 
definitely  required,  the  children  would  soon  learn  to  do  it 
properly. 

Of  course,  the  principal  question  here  is  whether  or  not  such 
employment  comes  properly  within  the  range  of  school  duties. 
Waiving  the  fact  that  nature  study  and  geography  rightly  call 
for  excursions;  is  it  not  the  purpose  of  the  school  to  bring  the 
pupil  into  an  understanding  and  appreciation  of  community  life, 
and,  also,  to  identify  him  with  that  life  as  much  as  possible,  in- 
cluding even  the  motor  action?  In  other  words,  is  not  action, 
"  doing  "  of  one  kind  or  another,  the  end-point  of  school  instruc- 
tion? Certainly  this  is  the  desired  end-point  in  literature,  in 


60  Teachers  College  Record  [150 

history,  and  in  manual  training;  and  if  it  is  likewise  true  of 
arithmetic,  might  not  this  be  one  of  the  most  effective  ways  for 
bringing  the  pupil  into  direct  touch  with  the  occupations  and  the 
people  about  him?  Often  such  a  task  as  that  suggested  above 
could  be  assigned  to  committees  of  the  class,  and  some  co- 
operative, administrative  ability  would  need  to  be  exercised 
before  proper  data  could  be  collected.  Why  not?  Even  the 
kindergarten  accepts  the  development  of  just  such  ability  as  one 
of  its  prominent  objects;  why  not  the  grades?  It  takes  time,  but 
it  is  worth  time.  If  the  school  is  to  cease  being  too  theoretical, 
it  must  find  in  some  manner  an  outlet  in  action.  If  data  for  live 
problems  must  be  collected  with  care,  it  is  only  natural  that  pupils 
do  some  of  this  work.  It  is  an  artificial  situation  when  nothing 
is  left  the  pupil  but  to  think  and  to  figure.  The  fact  that  the 
teacher  would  need  to  be  in  close  touch  with  the  environment  and 
reasonably  well  posted  as  to  the  facts,  in  order  to  carry  on  such 
work  successfully,  makes  it  all  the  more  desirable.  Whatever 
tends  to  make  our  teachers  less  "  bookish,"  and  to  identify  them 
with  their  surroundings,  particularly  in  a  quantitative  way,  is 
welcome. 

(d)  GENERAL  METHODS  OF  SOLVING  PROBLEMS 

(1)  General  Analysis.     By  this  we  mean  merely  common 
sense  applied  to  any  ordinary  problems.     Of  course  this  method 
is  of  the  highest  value.     It  excludes  the  memorizing  of  long  and 
ultra-scientific  forms:  of  analysis,  but  includes  a  brief  statement 
of  the  main  steps,  with  reasons. 

(2)  Unitary  Analysis.    A  special  form  of  analysis,  in  which 
a  series  of  terms  is  reduced  to  unity.     For  example :  If  two  men 
do  a  piece  of  work  in  4  days  of  10  hours  each,  how  many  men  will 
it  require  to  do  it  in  5  days  of  8  hours  each?    If  the  4  days  were 
each  i  hour  long,  it  would  take  20  men;  if  there  were  only  I  day, 
it  would  take  80  men;  but  since  there  are  5  days  it  will  take 
16   men,    and   since   the   days    are   8   hours   long,    it   will   take 
2  men. 

This  is  a  good  plan  for  solving  problems  which,  as  given  in 
arithmetics,  are  usually  of  a  bad  type.  For  example  the  anti- 
quated problems  of  compound  proportion  easily  yield  to  this 
method.  These  problems,  however,  are  bad  in  that  they  give  a 


151]  Work  of  the  Last  Three  Grades  61 

false  idea  of  business;  for  in  business  such  extensive  series  are 
very  rarely  met.  A  single  problem  here  corresponds  to  a  series 
of  problems  in  actual  life. 

(3)  The  equation  method.  This  is  the  latest  development 
in  the  science  of  arithmetic,  as  applied  to  daily  problems. 

The  equation,  as  needed  in  arithmetic,  rarely  involves  cases 
more  difficult  than  the  following,  where  a,  b,  and  c  stand  for 
known  numbers  :  — 

a  x  -\-  b  =  c, 


consequently  children  should  become  familiar  with  problems  like 
the  following:  — 

Twice  a  certain  number  is  58  ;  what  is  the  number  ? 

Five  times  a  certain  number,  together  with  6,  equals  41  ; 
what  is  the  number? 

Ten  plus  a  sixth  of  a  certain  number  equals  12  ;  what  is 
the  number? 

One-sixth  of  a  certain  number  equals  ^  ;  what  is  the 
number  ? 

Think  of  a  number  ;  multiply  it  by  5  ;  add  4  ;  tell  me  the  result 
and  I  will  tell  you  the  number. 

Problems  like  the  above,  while  applying  to  no  related  sub- 
ject, have  in  themselves  a  good  deal  of  interest  for  children. 
They  like  to  make  up  problems  like  the  last  one,  giving  them  to 
one  another  as  puzzle  games. 

This  subject  has  not  been  sufficiently  appreciated  by  teachers, 
but  it  is  of  highest  value;  for  example,  If  105^/2%  of  a  certain 
sum  is  $2,1  10,  what  is  the  sum?  Here,  1.055^  =  $2,110,  .'.x 
=  $2,000,  a  solution  much  simpler  than  the  conventional  one.  It 
is  evident  that  this  problem,  with  little  value  itself,  is  suggestive 
of  a  long  line  of  very  real  cases. 

(4)  Simple  proportion.  A  favorite  arithmetic  method,  under 
the  name  "  Rule  of  Three,"  before  the  equation  was  generally 
known.  It  is  merely  a  fractional  equation,  this  fact  being  concealed 
from  the  pupil  by  a  distinctive  symbolism.  As  an  instrument  of 
business  arithmetic  it  is  antiquated  and  cumbersome.  Its  value 
in  applied  arithmetic  lies  in  problems  in  physics,  as  in  the  pressure 
of  gases,  and  in  problems  involving  similar  figures.  Hence,  in 
ordinary  arithmetic  it  has  slight  value,  although  in  geometry  its 


62  Teachers  College  Record  [152 

value  is  very  great.    It  is  generally  better  to  translate  it  into  the 
simple  equation.    It  is  easier  to  solve 


2       20 

Or  2.QX  =  IO, 

than  to  solve 

20 :  5  =  2 :  x, 

although  all  these  expressions  are  equivalent. 
In  other  words,  a  proportion  like 

3:4=  15:* 
is  merely  equivalent  to 

*:i5  =  4:3> 
or  to 

x      4 

15       3 

which  is  solved  by  multiplying  by  15,  thus  avoiding  the  old  and 
ill-understood  proportion  rule. 

As  a  preparation  for  this  work  it  is  not  necessary  to  give  any 
elaborate  treatment  of  ratio.  The  concept,  ratio,  is  valuable  and 
necessary;  the  topic,  ratio,  is  not,  at  least  as  usually  treated. 

The  valuable  application  of  proportion  in  these  grades  is 
very  limited.  The  subject  might  be  applied  to  problems  like  this: 
If  $2  draw  $0.06  interest  in  i  year,  how  many  dollars  will  draw 
$3.50  interest  in  the  same  time?  But  in  the  first  place,  the  problem 
is  not  genuine;  and  furthermore,  if  it  were,  it  could  better  be 
solved  by  simple  analysis.  And  so  with  most  of  the  conventional 
problems  of  proportion.  There  is  one  subject,  however,  to  which 
proportion  necessarily  applies,  the  subject  of  similar  figures.  In 
as  far  as  these  have  an  interest  for  children,  the  subject  of  pro- 
portion is  valuable  in  the  grades.  In  particular,  problems  like  the 
following  are  recommended,  the  figure  always  being  drawn  in 
advance  of  the  solution: 

If  a  yard-stick,  standing  on  the  level  pavement,  casts  a 
shadaw  5  ft.  long,  and  the  shadow  of  the  gable  of  the  school 
building  at  the  same  time  is  145  ft.  long,  how  high  is  the  gable 
above  the  pavement  ? 


153]  Work  of  the  Last  Three  Grades  63 

The  following  form  of  solution  is  recommended: 

1 .  Let  x  =  the  number  of  feet  in  height. 

2.  Then  because  the  ratio  of  heights  equals  the  ratio  of  shadow 
lengths, 

x       145 


3.  Therefore, 

3  X  145 
x  = 


=  3  x  29 
=  87,  the  number  of  feet  required. 

(5)  Compound  proportion.  A  method  for  solving  certain 
obsolete  problems,  which  were  solved  in  the  i6th  and  I7th  cen- 
turies under  a  rule  known  by  various  names;  as  the  double  rule, 
the  rule  of  five,  the  rule  of  seven.  Practically  of  no  value. 

The  unreal  problems  commonly  found  in  text-books,  and  the 
entire  subject  of  compound  proportion,  should  be  omitted. 

(e)  THE  QUESTION  OF  GEOMETRY  IN  THE  GRADES 

With  the  exclusion  from  arithmetic  of  a  large  amount  of 
obsolete  matter,  which  had  gradually  been  accumulating  up  to 
within  a  few  years,  educators  looked  about  for  mathematical  sub- 
jects to  take  its  place.  The  most  convenient  material  at  hand  was 
algebra  and  geometry,  and  since  much  of  this  was  simpler  than 
what  had  been  excluded,  it  naturally  found  many  advocates.  As 
is  usual  in  such  cases,  the  movement  to  include  algebra  and 
geometry  in  the  grades  went  to  an  extreme,  with  the  result  that 
children  received  a  smattering  of  each  and  entered  the  high  school 
with  an  exaggerated  idea  of  their  knowledge  of  these  subjects 
or  with  a  preconceived  dislike  for  them.  The  question  now  begins 
to  assume  a  more  rational  form :  What  is  the  value  of  these  subjects 
in  the  grades,  and,  as  a  consequence,  how  should  they  be  presented  ? 
The  question  as  it  relates  to  algebra  has  already  been  briefly 
answered.  It  remains  to  consider  it  as  regards  geometry. 

Considering  only  the  ability  of  children  to  demonstrate  theo- 
rems or  to  solve  problems,  there  is  no  question  but  that  a  consider- 


64  Teachers  College  Record  [154 

able  amount  of  demonstrative  geometry  can  be  introduced  into 
the  last  three  of  the  eight  grades.  Considering  only  the  question 
of  valuable  mental  gymnastics,  the  same  answer  would  be  valid. 
But  if  the  subject  is  to  be  presented  in  the  same  way  as  in  the 
high  school,  and  if  the  child  is  there  to  go  over  the  same  ground, 
the  effect  upon  him  is  not  altogether  happy.  It  is  a  common 
remark  of  teachers  of  geometry  that  they  would  prefer  a  pupil 
who  had  never  studied  geometry  to  one  who  had  taken  it  in  the 
grammar  grades.  The  edge  of  interest  is  worn  off,  because 
nothing  new  is  being  mastered. 

If,  then,  the  subject  is  to  be  taken  in  the  grades,  it  should 
be  in  such  way  as  not  only  to  appeal  to  the  pupil's  interests  then, 
but  to  leave  that  impression  of  only  partial  mastery  which  acts  as 
a  stimulus  to  further  consideration  of  the  subject.  What,  then, 
are  these  interests,  and  how  shall  they  be  met? 

In  connection  with  the  pupil's  mathematical  work  it  is  con- 
venient for  him  to  solve  an  equation.  Therefore  the  equation 
is  introduced  into  his  mathematics ;  not  algebra  as  a  separate  topic, 
but  such  features  of  algebra  as  bear  upon  the  work  in  hand. 
Likewise  it  is  convenient  for  him  to  measure  things.  Therefore 
portions  of  geometry  should  come  into  his  mathematics ;  not 
geometry  as  a  separate  science,  but  such  features  of  it  as  bear 
upon  the  work  in  hand,  and  with  the  increase  of  a  rational  manual 
training  this  work  is  daily  becoming  more  important. 

This  work  in  mensuration  requires  that  the  pupil  should  know 
the  names  of  the  common  surfaces  and  solids.  This  part  of 
geometry,  therefore,  is  introduced  in  connection  with  mensuration. 
It  is  also  necessary  that  he  should  know  how  to  measure  these 
common  forms ;  hence,  this  work  is  also  taken  up,  and  in  a  manner 
quite  scientific  enough  to  merit  the  name  geometric.  As  to  the 
actual  geometric  facts  absorbed  by  the  pupil,  this  has  been  done 
for  years.  The  new  feature,  developed  of  late  years,  is  the  dis- 
covery of  the  facts.  Formerly  it  was  merely  to  memorize  a  set 
of  rules  dogmatically  given;  now  it  is  to  work  these  rules  out. 

Bibliography:  Smith's  Teaching  of  Elementary  Mathematics, 
as  already  noted;  Beman  &  Smith's  Higher  Arithmetic,  much  of 
which  is  suggestive  for  grade  work,  both  in  business  problems 
and  in  mensuration ;  on  the  geometry,  consult  Sundara  Row,  Geo- 
metric Paper  Folding,  much  of  which  is  suggestive  for  grade 
teachers,  although  the  book  is  not  entirely  elementary. 


IV.     OUTLINE  FOR  THE  LAST  THREE  GRADES 

Grade  VI 

In  this  grade  the  reduction  of  common  fractions  to  decimals, 
and  vice  versa,  is  the  only  topic  in  pure  arithmetic  demanding 
attention;  except  as  the  others  enter  into  reviews.  The  year  is 
given  to  applications,  chiefly  in  percentage  and  denominate  num- 
bers. 

At  the  beginning  of  the  year  the  work  of  Grade  V  should 
be  reviewed  as  suggested  in  the  notes  for  that  grade.  On  the 
subject  of  reviews  in  general,  and  the  use  of  the  text-book,  con- 
sult the  general  suggestions  of  the  preceding  grades.  It  is  par- 
ticularly desirable  to  carry  out  the  general  policies  as  laid  down 
in  the  notes  on  the  work  of  all  preceding  grades,  and  hence 
teachers  and  those  carrying  on  professional  observation  work 
should  become  familiar  with  those  notes. 

The  following  are  some  of  the  topics  especially  suitable  for 
the  sixth  school  grade:  I.  Gardening;  2.  Farming;  3.  Rainfall, 
variations  in  different  places  and  from  year  to  year;  4.  Mining; 
5.  Lumbering;  6.  Fishing;  7.  Ranching;  8.  Transportation;  9. 
Manufacture  and  trade,  introducing  the  metric  system  as  con- 
nected with  our  rapidly  growing  foreign  market. 

Aside  from  the  English-speaking  world,  the  metric  system 
is  generally  used  in  highly  civilized  countries.  Our  exports  of 
guns,  machinery,  and  all  manufactured  articles  are  dependent  to 
quite  a  degree  upon  the  use  of  metric  measures  in  all  descriptions 
of  our  goods.  The  foreign  trade  cannot  be  expected  to  sur- 
render its  extremely  simple  system  to  our  cumbersome  one;  and 
whether  we  adopt  the  metric  measures  for  our  own  use  or  not, 
we  must  adopt  them  for  our  export  trade,  if  we  wish  to  see  it 
developed. 

The  traditional  topics  especially  considered  this  year  are :  ( I ) 
Simple  commercial  transactions;  bills,  receipts,  commercial  dis- 
counts 5(2)  Simple  investments ;  notes,  mortgages,  simple  interest. 
'55]  65 


66  Teachers  College  Record  [156 

The  drill  with  abstract  numbers  should  always  hereafter 
include  the  important  problem  of  the  addition  of  columns  of 
figures,  checking  by  adding  in  reverse  order.  It  should  also 
include  subtraction,  generally  of  numbers  not  exceeding  thou- 
sands; multiplication  of  numbers  like  13  X  I2j^,  all  these  being 
cases  frequently  met  in  ordinary  business ;  and  division  with  one- 
figure  or  two-figure  divisors.  In  general  all  this  work  should 
relate  to  the  problems  of  daily  life,  rather  than  to  tedious  and 
artificially  difficult  problems  intended  merely  as  tasks. 

In  geometry,  the  work  is  confined  to  the  measurement  of 
surfaces  and  of  solids,  as  indicated  below;  only  commensurable 
magnitudes  (for  example,  lines  having  a  common  measure) 
being  considered.  Paper- folding  (Sundara  Row,  Geometric  Exer- 
cises in  Paper  Folding,  pages  1-7,  14-15),  paper-cutting,  and  draw- 
ing, should  be  employed  to  illustrate  mensuration  propositions  as 
in  the  earlier  grades.  The  proofs  of  the  following  propositions, 
for  commensurable  lines,  are  entirely  within  the  powers  of  pupils 
of  this  grade: 

The  area  of  a  square  a  units  on  a  side,  is  a2  square  units. 

The  area  of  a  rectangle,  a  units  high,  b  units  long,  is  ab  square 
units. 

The  area  of  a  parallelogram  a  units  high,  b  units  long,  is  ab 
square  units. 

The  area  of  a  triangle,  a  units  high,  b  units  long,  is  #ab 
square  units. 

This  work  is  a  proper  part  of  mathematics  for  children  of  this 
age.  The  mathematical  unity  is  preserved;  the  small  amount  of 
geometry  not  being  looked  upon  as  a  separate  study.  For  the 
sixth  grade,  besides  the  work  above  specified,  the  following  should 
be  included:  Mensuration  of  the  circle,  cylinder,  and  the  forms 
above  mentioned;  board  measure  (foot  and  thousand)  and  cord 
measure ;  application  to  manual  training  and  to  building. 

The  following  problems  are  suggested  as  a  type  of  a  series 
devoted  to  a  single  subject: 

1.  A  township  in  the  western  farming  states  is  usually  6  miles  square. 
How  many  square  miles  does  it  contain?    Show  by  a  drawing. 

2.  One  square  mile  contains  640  acres  of  land  and  is  often  divided 
into  four  equal  parts,  called  quarter-sections,  for  four  farms.    What  is  the 
size  of  each  farm?     What  is  the  distance  around  each  farm?     Show  by 
a   drawing. 

3.  Divide  160  acres  into  4  equal  squares.    How  many  acres  in  each? 


157]  Outline  for  the  Last  Three  Grades  67 

How  far  is  it  around  each?     Show  by  a  drawing.     How  long  would  it 
take  you  to  walk  around  each? 

4.  Divide  40  acres  into  4  equal  squares.     How  many  acres  in  each? 
What  is  the  length,  in  feet,  of  one  of  its  sides?    The  distance  around  it 
is  what  part  of  I  mile? 

5.  Compare  the  area  of  some  city  block  near  you,  with  a  10  acre  plot 
of  ground.    Make  drawings  to  a  scale. 

6.  One  acre  contains   160  square  rods,  or  is  equal  to  a  square  with 
each  side  12.65  rods  long.     Mark  out  such  an  area  somewhere  near  the 
school.     How  many  feet  around  it? 

7.  How   many   acres   in   the   block   in   which   your   school   building 
stands?    Pace  it  off,  taking  2  ft.  to  a  pace,  in  order  to  measure  it. 

8.  The  average  farm  in  the  United  States  contains  about  140  acres. 
That  is  what  part  of  one  square  mile?    What  part  of  one  quarter  section? 

9.  An  ordinary  city  lot,  25  by  100  feet,  is  what  part  of  an  acre? 

10.  The  land  on   which  your   school-building  stands,   including  the 
yard,  is  what  part  of  an  acre?     Estimate  the  area  of  other  portions  of 
land,  and  test  by  measurement  to  see  how  nearly  right  you  are. 

11.  Good  farm  land  in  New  York  State  can  often  be  bought  for  about 
$60  per  acre.    According  to  that,  what  would  be  the  price  of  the  block  of 
land  in  which  your  school  building  stands?     Find  out  the  approximate 
worth  of  this  plot  of  land?     Give   some  reasons   for  the  difference  in 
price,  if  there  is  a  difference. 

12.  Plot  out  a  40  acre  piece  of  land  in  your  vicinity.     Also  one  of 
160  acres.     Show  that  you  are  approximately  right. 

13.  A  certain  farmer  in  Central  Ohio  has  a  farm  of  160  acres.    Show 
a  plot  of  this  ground,  drawn  on  the  scale  of  Vie  of  one  inch  to  one  rod. 

14.  Mark  off  a  suitable  area  for  this  man's  orchard,  yard  and  garden, 
barn  and  barn-yard,  field  for  corn,  field  for  clover  and  hay,  field  for  wheat, 
one  for  pasture,  and  a  timber  lot. 

15.  It  takes  him  about  54  of  a  day  to  prepare  an  acre  of  land  for 
planting  corn.    How  long  would  it  take  to  prepare  the  field  that  you  have 
marked  out  for  corn? 

16.  I  takes  him  about  J4  of  a  day  to  plant  an  acre  of  corn,  and  the  same 
to  sow  an  acre  of  wheat.   How  long  would  it  take  to  plant  and  sow  these 
two  fields? 

17.  It  takes  a  peck  of  corn  to  plant  an  acre,  and  1^4  bushel  of  wheat 
to  sow  an  acre.    How  much  corn  and  wheat  are  necessary  for  your  fields? 

18.  The  average  yield  of  corn  per  acre  on  this  man's  farm  is  about 
40  bushels.     How  much  corn  would  the  above  field  yield? 

19.  The  average  yield  of  wheat  per  acre  is  about  18  bushels.     How 
much  wheat  would  the  above  field  yield? 

20.  The  highest  prices  that  this  farmer  has  received  are  50  cents  a 
bushel  for  corn  and  80  cents  a  bushel  for  wheat.    What  facts  can  you  dis- 
cover about  the  possible  income  of  the  farmer? 

Such  a  series  suggests  how  arithmetic  in  the  higher  grades 
might  give  an  extensive  knowledge  concerning  an  occupation, 


68  Teachers  College  Record  [158 

and  at  the  same  time  furnish  abundant  drill  in  pure  arithmetical 
work. 

Grade  VII 

Review  the  work  of  Grade  VI  as  suggested  by  a  similar  note 
under  that  grade. 

The  following  topics  in  business  arithmetic  constitute  a  large 
part  of  this  year's  work:  Commercial  discount;  per  cent  of  gain 
or  loss;  commission  and  brokerage;  taxes  with  customs,  treated 
as  suggested  on  p.  56 ;  simple  interest ;  compound  interest  as  used 
in  savings  banks. 

The  spirit  in  which  this  work  is  treated  is  suggested  in  the 
notes  on  Grade  VI.  See  Beman  and  Smith's  Higher  Arithmetic, 
the  portion  relating  to  business  arithmetic  being  entirely  suited 
to  seventh  and  eighth-grade  work. 

The  nature  of  the  oral  drill  is  set  forth  on  p.  41. 

Mensuration  and  geometry :  Review  mensuration  of  the  circle 
and  cylinder ;  add  the  pyramid  and  cone ;  in  connection  with  this, 
review  the  geometric  knowledge  thus  far  secured,  covering  the 
ground  laid  down  in  Hanus,  Geometry  in  the  Grammar  School, 
pages  37-46. 

Ratio  and  proportion:  See  p.  61. 

Grade  VIII 

i.  General  Suggestions 

See  p.  65.  Children  have  now  become  mature  enough  to 
appreciate  difficult  business  transactions,  and  these,  together  with 
certain  geometry  work,  constitute  this  year's  work. 

2.  The  Mathematical  Work 

(a)  Corporations :  Organization,  stocks  and  dividends,  bonds 
and  interest. 

For  this  work,  the  best  text-book  is  the  financial  page  of  a 
daily  newspaper.  It  is  not  necessary  to  enter  into  all  of  the  cus- 
toms of  the  stock  exchange;  indeed  only  a  few  technical  ex- 
pressions are  of  any  use  to  the  average  citizen.  But  a  dozen 
problems  involving  the  cost  of  ten  shares  of  various  specified 
stocks,  and  another  dozen  involving  the  amounts  to  be  received 


159]  Outline  for  the  Last  Three  Grades  69 

from  the  sale  of  certain  stocks,  and  a  few  relating  to  the  pur- 
chase of  bonds,  all  of  the  problems  being  based  on  the  news- 
paper quotations  and  all  involving  the  usual  brokerage,  will  be 
interesting  and  valuable.  Different  pupils  will  have  different 
answers,  depending  on  the  quotations  used,  which  brings  up  the 
questions  of  the  meaning  of  the  expressions  "  highest,"  "  lowest," 
"  opening,"  and  "  closing,"  and  of  the  causes  of  fluctuation  from 
day  to  day.  (See  also  p.  55.) 

(&)  Lending  money:  Notes,  bonds,  and  mortgages;  partial 
payments  of  a  practical  nature;  writing  notes  and  receipts. 

(c)  Banking :  Starting  accounts ;  bank  books,  deposit-slips, 
checks,  the  blanks  being  in  actual  use;  bank  discount;  savings 
banks  and  compound  interest. 

It  is  now  possible  to  procure  school  sets  of  blanks,  including 
bank  book,  deposit  slips,  checks,  notes,  and  drafts.  The  use  of 
such  sets  adds  greatly  to  the  interest  and  the  value  of  the  work. 

(d)  Exchange  treated  from  the  modern  standpoint;  drafts, 
checks,  money-orders. 

The  exchange  rates  of  banks  may  be  found  on  the  financial 
page  of  any  of  the  leading  daily  papers. 

(e)  Our   foreign   trade;   its   growth,   customs,   duties;  the 
metric  system  reviewed,  with  the  story  of  its  origin,  the  extent 
of  its  use,  its  advantages ;  foreign  exchange  and  the  money  systems 
of  England,  France  and  Germany;  longitude  and  time  reviewed 
in  connection  with  study  of  geography  of  the  countries  involved. 

(/)  Insurance. 

(g)  Mensuration  and  geometry:  Review  mensuration  of  im- 
portant figures,  and  cover  the  work  laid  down  in  Hanus,  Geometry 
in  the  Grammar  Grades,  part  2,  page  47-52.  In  connection  with 
this  work,  consider  square  root  as  stated  below. 

(h)  Square  root  approached  through  practical  problems: 
Relation  between  the  geometric  and  the  algebraic  forms ;  the  latter 
used  in  explaining  the  process. 

Since  this  article  treats  mainly  of  arithmetic  in  the  grades, 
merely  suggesting  the  use  of  algebra  and  geometry ;  and  since  in 
the  eighth  grade  it  is  intended  to  review  the  arithmetic  with  special 
reference  to  its  business  applications,  and  to  devote  some  time  to 
elementary  algebra  and  geometry,  further  consideration  of  the 
latter  work  is  reserved  for  a  subsequent  article. 

In  general  it  may  be  said,  however,  that  there  is  a  manifest 


/o  Teachers  College  Record  [160 

danger  of  overdoing  the  work  in  algebra  and  geometry  in  the 
eighth  grade.  Except  so  far  as  these  subjects  have  practical 
application  to  problems  of  the  life  or  the  science  that  the  pupils 
of  this  grade  meet  or  have  interest  in,  the  work  may  better  be 
left  for  the  high  school.  Therefore  the  work  in  geometry  may 
well  be  limited  to  mensuration  and  constructions,  and  that  in 
algebra  may  well  be  directed  to  the  applications  of  simple  equa- 
tions. Any  elaborate  treatment  of  algebraic  functions,  such  as 
complicated  fractions,  or  any  considerable  amount  of  work  in 
literal  equations,  is  of  doubtful  value  at  this  time.  Teachers  who 
contribute  to  the  stock  of  genuine  applications  of  the  simple 
equation,  especially  in  the  problems  of  business  or  of  simple 
science,  will  perform  a  real  service  to  education,  for  it  is  here 
that  the  greatest  value  of  algebra  for  the  eighth  grade  is  to  be 
found. 


Can   Card  Games  be  Used  to 
Advantage  in  the  Schoolroom  ? 

THIS  is  a  question  which  teachers  everywhere  are  asking.     Card  Games,  prepared 
by  practical  schoolmen,  are  being  advertised  in  this  and  other  educational  jour- 
nals.    They  are  constructed  and  edited  by  educators  of  note.     The  claim  is  made 
by  the  publishers,  and  with  obviously  good  reason,  that  these  games  interest  the  pupil ; 
stimulate  him  to  greater  efforts  and  secure  better  results  than  are  possible  with  usual 
and  routine  methods  alone. 

Should  these  claims  be  justified  by  actual  schoolroom  use,  then  teachers  everywhere 
who  wish  to  keep  abreast  of  the  times  will  be  glad  to  know  it,  and  will  be  interested  in 
the  following  reports  from  teachers  who  are  using  these  games. 

Dr.   A.    E.   WINSHIP,   Editor   "  Journal   of   Education,"   Boston,   Mass. 

"  It    is   in    every    way   interesting,    is    easily    learned,    and    makes   the    children    quick    in    their 
combinations.      It  is  ingenious  and  attractive.  " 
JONATHON    RIGDOV,    President   Central    Normal   College,    Danville,    Ind. 

"  I   am   glad   to   assure    you   that   I    tried   your    game    of   Addition   and    Subtraction   and   most 
heartily  recommend  it.   It  seems  to  me  that  it  is  not  possible  to  add  more  pleasure  to  a  game.     I 
shall  be  glad  to  speak  to  our  students  concerning  each  of  these  games  that  I  have  an  opportunity 
to  test.     If  your  Multiplication  and  Division  game  is  ready,  please  send  it." 
Mr.   E.  J.   LLEWELYN,  City  Supt.,  Arcadia,  Ind. 

"  While  we  have  not  had  time  to  put  the  games  to  a  very  severe  test  I  can  truthfully  say 
that  I  believe  them  to  be  all  and  more  than  represented  to  be.  In  the  short  time  that  we  have  had 
the  game,  we  have  had  it  in  almost  constant  use  in  our  3rd  and  4th  grades,  and  are  much  pleased 
with  the  results.  I  believe  that  such  games  as  these  will  have  a  tendency  to  awaken  an  interest 
among  students  in  the  study  of  Arithmetic.  It  quickens  their  ability  to  add  and  subtract  readily 
and  accurately  and  exerts  a  good  moral  influence  over  the  pupils.  I  have  no  criticism  to  offer. 
They  are  just  the  thing." 
W.  A.  COLLINGS,  City  Supt.,  Charlestown,  Ind. 

"  We  are   greatly   pleased   with   the   results   of   the    Mathematical   games.      They   present   a   dry 
subject   in  an   attractive   manner  and   hold   the   child's  attention   without  effort." 
J.    B.    JORDAN,    Principal,    Carbon,    Texas. 

"  I  find  the  Fraction  games  not  only  interesting  and  fascinating,  but  instructive,  and  am 
sure  they  will  be  a  means  to  good  results  in  the  way  of  mental  development  and  individual 
improvement.  I  have  found  the  Addition  game  very  interesting  for  the  children  in  class  use. 
It  is  of  special  benefit  to  those  who  have  little  interest  in  arithmetic  generally,  in  the  way  of 
arousing  an  interest  in  class  work." 
Mr.  HORACE  ELLIS,  President  Idaho  State  Normal  School,  Allison,  Idaho. 

"  We    tried    this    game    with    some    beginning    preparatory    students.      Results    were    interesting 
and   highly   valuable.   We   believe   the   game   strictly   pedagogic   and   valuable." 
Mr.    PAUL   A.    COWGILL.    City    Supt,   Michigan    City,    Ind. 

"  I   have  tried  the  addition    and  subtraction  game    with  pupils  selected   from  the  4th,    sth   and 
6th   grades,   and   find  they  are  intensely   interested   in  it.      It  is   undoubtedly   of   great   educational 
value.     I   am  sure  that  it  is  just  what   we  need." 
Mr.    W.    E.    STIFF,    County    Supt.,    Mitchell,    Ind. 

"  I  have  tested  the  game   of  addition  and  subtraction  and   find  it  to  be   very  interesting.      It 
is  a  game  that  gives  the  child  a  valuable  drill  in  learning  the  necessary  combinations  of  numbers." 
Mr.    E.    C.    SHIELDS,   County   Supt.,    Clearfield,    Pa. 

"  I  find  the  game  very  instructive  and  beneficial  in  rapid  and  accurate  operations  in  addition 
and  subtraction.     They  would  be  a  good  thing  in  any  school  or   home." 
Mr.    I.    V.    BUSBY,   Ex.    Supt.,    City   Schools,   Alexandria,   Ind. 

"  It  is  highly  _interesting,  and  helpful  in  a  marked  degree  in  stimulating  rapid  combinations 
of  numbers.  I  believe  the  games  will  prove  to  be  a  happy  device  for  strengthening  pupils  in  their 
number  operations." 

Mr.  H.   F.  GALLIMORE,  City  Supt,  Zionsville,  Ind. 

"  I    placed    the    game   of   addition   and    subtraction    in   the   hands   of   one    of   my   teachers   and 
instructed    her    to    try    it    with    the    children.      She    reports    that    the    results    are    very    gratifying. 
I   feel  that  you  are  making  a  valuable  addition  to  the  equipment  of  our  schools." 
Mr.  (G-RANT    GOSSETT,    County    Supt.,    Covington,    Ind. 

"  I  found  it  very  instructive  and  beneficial  because  it  is  attractive  and  interesting  to  the 
children. 

A  sample  of  either  "Multiplication  and  Division,"  "Addition  and 
Subtraction, "  or  "  Fractions,"  will  be  sent  postpaid  on  receipt  of 
25  cents. 

THE    CINCINNATI    GAME    CO. 

Cincinnati,   Ohio 

71 


THE  TEACHING  OF  .... 
ELEMENTARY  MATHEMATICS 

By  DAVID  EUGENE  SMITH 

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teachers  who  wish  to  know  something  of  these  great  questions 
of  teaching.  Certain  methods  are  suggested  that  can  be  applied 
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works  which  the  author  has  found  of  great  value. 

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aggregating  $11,313,890.   Send  for  one-hundred  page  manual  giving  list  of  positions  filled. 

H.  E.  CROCKER,      ) 

New  York  Office:  156  Fifth  Ave.  W.  D.  KERR,  >  Managers. 

Tel.  No.  3971-8-18  st.  P.  V.  HUYSSOON,   ) 


ALBANY  TEACHERS'  AGENCY 

Provides  schools  of  all  grades  with  competent  teachers 
Assists  teachers  in  obtaining  positions 

HARLAN  P.  FRENCH,  Proprietor.    £/  Chapel  Street,  ALBANY.  N.  Y. 


7O  Fifth  Avenue 
••  N —  York 


fl?  Pratt  Teachers'  Agency 

Recommends  college  and  normal  graduates,  specialists,  and  other  teachers 
to  colleges,  public  and  private  schools,  and  families. 

Advises  parents  about  schools.  WM.  O.  PRATT,  Manager 


77 


Seacbers  College 
Columbia 


Teachers  College  is  the  professional  school  of  Columbia  University  for  the  study 
of  education  and  the  training  of  teachers.  The  purpose  of  the  College  is  to  afford 
opportunity,  both  theoretical  and  practical,  for  the  training  of  teachers  of  both  sexes 
for  elementary,  secondary,  and  normal  schools,  of  specialists  in  various  branches  of 
school  work,  and  of  principals,  supervisors,  and  superintendents  of  schools. 

The  College  offers  56  courses  in  Education,  including  6  courses  on  the  History 
and  Principles  of  Education,  4  courses  on  Educational  Administration,  7  courses  on 

Educational  Psychology,  and  25  courses  on  the  theory  and  practice 
Courses  of  teaching  Biology,  Domestic  Art,  Domestic  Science,  English,  Fine 

of  Arts,   French,  Geography,   German,  Greek,   History,   Kindergarten, 

Instruction  Latin,  Manual  Training,  Mathematics,  Music,  Physical  Science  and 

Physical  Education.  Other  courses  of  instruction  supplementary 
to  those  above  are  as  follows :  Biology,  6  courses ;  Domestic  Art,  5  courses ;  Domestic 
Science,  10  courses ;  English,  6  courses ;  Fine  Arts,  14  courses ;  French,  3  courses ; 
German,  2  courses ;  Geography,  4  courses ;  History,  4  courses ;  Kindergarten,  4  courses ; 
Manual  Training,  9  courses ;  Mathematics,  3  courses ;  Music,  5  courses ;  Physical 
Science,  4  courses,  and  Physical  Education,  6  courses.  Qualified  students  of  Teachers 
College  may  also  pursue  University  courses  in  History,  Language  and  Literature, 
Natural  Science,  Mathematics,  Philosophy,  Psychology,  Ethics,  Anthropology,  Music, 

Economics  and  Social  Science.  Teachers  College  maintains  two 
Teachers  schools  of  observation  and  practice:  one,  the  Horace  Mann  School, 

College  the  other  known  as  the  Speyer  School.    The  Horace  Mann  School 

Schools  comprises  three  departments  —  a  kindergarten  for  children  of  three 

to  six  years  of  age,  an  elementary  school  of  eight  grades,  and  a 
high  school  of  four  grades.  The  Speyer  School  consists  of  a  kindergarten,  elementary 
school,  and  special  classes  in  sewing,  cooking  and  manual  training. 

Courses  of  Study  are  as  follows:  (i)  A  two-year  Collegiate 
Courses  Course  which  if  followed  by  a  two-year  professional  course  leads 

of  to  the  degree  of  B.S. ;    (2)    Two-year  professional   courses   lead- 

Study  ing  to   the   Bachelor's   diploma   in    (a)    Secondary   Teaching,    (b) 

Elementary  Teaching,  (c)  Kindergarten,  (d)  Domestic  Art,  (e) 
Domestic  Science,  (f)  Fine  Arts,  (g)  Music,  and  (h)  Manual  Training;  (3)  Gradu- 
ate courses  of  one  and  two  years,  respectively,  leading  to  the  Master's  and  Doctor's 
diplomas  in  the  several  departments  of  the  College.  Students  holding  the  degree  of 
B.S.  or  A.B.  may  become  candidates  for  A.M.  and  Ph.D. 

The  requirements  for  admission  are  as  follows:  (i)  To  the  Col- 
Admission  legiate  Course  —  completion  of  a  high-school  course;  (2)  to  the 
Require-  two-year  courses —  (a,  b,  and  c  above)  completion  of  the  Collegiate 
ments  Course  or  its  equivalent  in  an  approved  college  or  graduation  from 

an  approved  normal  school;  (d,  e,  f,  g,  h)  same  as  for  (a)  and  (fc) 
or  two  years  of  technical  training  or  experience  in  teaching;  (3)  to  the  graduate  courses 
—  college  graduation  or  its  equivalent. 
p  ..         .  .  Tuition  in  graduate  courses  and  courses  leading  to  a  degree,  $150; 

in  other  courses,  $100.    The  faculty  annually  awards  5  Fellowships 

of  $650  each,  i  Scholarship  of  $400,  12  Scholarships  of  $150  each, 
Scholarships  and  4  Scholarships  of  $75  each. 

For  circulars  and  further  information,  address  the  Secretary. 

JAMES   E.   RUSSELL,   Ph.D.,    Dean. 

78 


Columbia  "University 
in  tbe  City  of  IRew  li>orfc 

Columbia  University  includes  both  a  college  and  a  university  in  the  strict  sense 
of  the  words.  The  college  is  Columbia  College,  founded  in  1754  as  King's  College. 
The  University  consists  of  the  Faculties  of  Law,  Medicine,  Philosophy,  Political  Science, 
Pure  Science,  and  Applied  Science. 

The  point  of  contact  between  the  college  and  the  university  is  the  senior  year 
of  the  college,  during  which  year  students  in  the  college  pursue  their  studies,  with 
the  consent  of  the  college  faculty,  under  one  or  more  of  the  faculties  of  the  university. 

Barnard  College,  a  college  for  women,  is  financially  a  separate  corporation;  but, 
educationally,  is  a  part  of  the  system  of  Columbia  University. 

Teachers  College,  a  professional  school  for  teachers,  is  also,  financially,  a  sep- 
arate corporation ;  and  also,  educationally,  a  part  of  the  system  of  Columbia  University. 

Each  college  and  school  is  under  the  charge  of  its  own  faculty,  except  that  the 
Schools  of  Mines,  Chemistry,  Engineering,  and  Architecture  are  all  under  the  charge 
of  the  Faculty  of  Applied  Science. 

For  the  care  and  advancement  of  the  general  interests  of  the  university  educa- 
tional system,  as  a  whole,  a  Council  has  been  established,  which  is  representative  of 
all  the  corporations  concerned. 


I.    THE  COLLEGES. 

Columbia  College  offers  for  men  a  course 
of  four  years,  leading  to  the  degree  of 
Bachelor  of  Arts.  Candidates  for  admission 
to  the  college  must  be  at  least  fifteen  years 
of  age,  and  pass  an  examination  on  pre- 
scribed subjects,  the  particulars  concerning 
which  may  be  found  in  the  annual  Circular 
of  Information. 

Barnard  College,  founded  in  1889,  offers 
for  women  a  course  of  four  years,  leading 
to  the  degree  of  Bachelor  of  Arts.  Candi- 
dates for  admission  to  the  college  must  be 
at  least  fifteen  years  of  age,  and  pass  an 
examination  on  prescribed  subjects,  the 
particulars  concerning  which  may  be  found 
in  the  annual  Circular  of  Information. 

II.    THE  UNIVERSITY. 

In  a  technical  sense,  the  Faculties  of  Law, 
Medicine,  Philosophy,  Political  Science, 
Pure  Science,  and  Applied  Science,  taken 
together  constitute  the  university.  These 
faculties  offer  advanced  courses  of  study 
and  investigation,  respectively,  in  (a)  pri- 
vate or  municipal  law,  (b)  medicine,  (c) 
philosophy,  philology,  and  letters,  (d)  his- 
tory, economics,  and  public  law,  (e)  mathe- 
matics and  natural  science,  and  (/)  applied 
science.  Courses  of  study  under  all  of 
these  faculties  are  open  to  members  of  the 
senior  class  in  Columbia  College.  Certain 
courses  under  the  non-professional  faculties 
are  open  to  women  who  have  taken  the  first 
degree.  ^  These  courses  lead,  through  the 
Bachelor's  degree,  to  the  university  degrees 
of  Master  of  Arts  and  Doctor  of  Philosophy. 
The  degree  of  Master  of  Laws  is  also  con- 
ferred for  advanced  work  in  law  done  under 
the  Faculties  of  Law  and  Political  Science 
together. 

III.    THE  PROFESSIONAL  SCHOOLS. 

The  Faculties  of  Law,  Medicine,  and  Ap- 
plied Science  conduct  respectively  the  pro- 
fessional schools  of  Law,  Medicine,  and 
Mines,  Chemistry,  Engineering,  and  Archi- 


tecture, to  which  students  are  admitted  as 
candidates  for  professional  degrees  on  terms 
prescribed  by  the  faculties  concerned.  The 
faculty  of  Teachers  College  conducts  profes- 
sional courses  for  teachers,  that  lead  to  a 
diploma  of  the  university. 

1.  THE   SCHOOL   OF   LAW,   established   in 
1858,  offers  a  course  of  three  years,  in  the 
principles  and  practice  of  private  and  public 
law,  leading  to  the  degree  of  Bachelor  of 
Laws. 

2.  THE     COLLEGE    OF     PHYSICIANS    AND 
SURGEONS,  founded  in  1807,  offers  a  course 
of  four  years,  in  the  principles  and  practice 
of  medicine  and  surgery,  leading  to  the  de- 
gree of  Doctor  of  Medicine. 

3.  THE  SCHOOL  OF  MINES,  established  in 
1864,  offers  courses  of  study,  each  of  four 
years,   leading  to  a  professional   degree   in 
mining  engineering  and  in  metallurgy. 

4.  THE   SCHOOLS   OF   CHEMISTRY,    ENGI- 
NEERING, AND  ARCHITECTURE,  set  off  from 
the  School  of  Mines  in  1896,  offer  respect- 
ively, courses  of  study,  each  of  four  years, 
leading  to   an   appropriate   professional    de- 
gree, in  analytical  and  applied  chemistry ;  in 
civil,  sanitary,  electrical,  and  mechanical  en- 
gineering; and  in  architecture. 

5.  TEACHERS   COLLEGE,   founded  in    1888 
and  chartered  in  1889,  was  included  in  the 
university  in   1898.     It  offers  the  following 
courses  of  study :  (a)  graduate  courses  lead- 
ing to  the  Master's  and  Doctor's  diplomas 
in  the  several  departments  of  the  College ; 
(b)  professional  courses,  each  of  two  years, 
leading  to  the  Bachelor's  diploma  for  Sec- 
ondary    Teaching,     Elementary     Teaching, 
Kindergarten,      Domestic      Art,      Domestic 
Science,    Fine    Arts,    Music,    and    Manual 
Training;    (c)    a   collegiate   course   of   two 
years,    which,    if    followed    by    a    two-year 
professional  course,  leads  to  the  degree  of 
Bachelor  of  Science.     Certain  of  its  courses 
may    be    taken,    without    extra    charge,    by 
students  of  the  university  in  partial  fulfil- 
ment  of  the   requirements   for  the   degrees 
of  Bachelor  of  Arts,   Master  of  Arts,   and 
Doctor  of   Philosophy. 

NICHOLAS  MURRAY  BUTLER,  LL.D., 

President. 


79] 


Teachers  College  Record 


No.  x 
January 


No.  a 
March 

No.  3 
May 

No.  4 
September 

No.  5 

November 


CONTENTS  OF  VOLUME  I  — 1900 

The  Function  of  the  University  in  the  Training  of  Teachers.  —  JAMES 

E.  RUSSELL. 
Historical  Sketch  of  Teachers  College  from  its  Foundation  to   1897. 

—  WALTER  L.  HERVEY. 
The   Organisation  and  Administration   of   Teachers  College.  —  JAMES 

E.  RUSSELL. 

Aims  of  Nature  Study.  —  FRANCIS  E.  LLOYD. 

Outline  of  Course  in  Nature  Study  in  the  Horace  Mann  School,  etc. — 
ELIZABETH  CARSS. 

Outline  of  Course  in  English  in  the  Horace  Mann  School.  —  FRANKLIN 

T.  BAKER. 
English  Composition  and  Topical  Studies  in  Literature.  —  HERBERT  V. 

ABBOTT,  ELLEN  Y.   STEVENS,  and  EMILY  BRINCKERHOFF. 

Syllabi  for  Teachers  College  Courses: 

History  of  Education.  —  PAUL  MONROE.  Principles  of  Education.  — 
NICHOLAS  MURRAY  BUTLER.  School  Administration.  —  SAMUEL  T. 
BUTTON.  National  Educational  Systems.  —  JAMES  E.  RUSSELL. 

Outlines  of  Courses  in  Hand  Work  in  the  Horace  Mann  School: 
Fine    Arts.  —  ALFRED    V.    CHURCHILL.      Domestic    Art.  —  MARY    S. 
WOOLMAN.     Domestic  Science.  —  HELEN  KINNE.     Manual  Train- 
ing. —  CHARLES  R.  RICHARDS. 


CONTENTS  OF  VOLUME  II  — 1901 

No.  i  Biology  in  the  Horace  Mann  High  School.  —  FRANCIS  E.  LLOYD  and 

January  MAURICE  A.  BIGELOW. 

No.  a  Geography  in  the  Horace  Mann  Schools.  —  RICHARD  E.  DODGE  and  Miss 

March  C.  B.  KIRCHWEY. 

No.  3  Child  Study.  —  Sources  of  Material  and  Syllabi  of  College  Courses.  — 

May  EDWARD  L.  THORNDIKE. 

No.  4  Syllabi  of  Courses  in  Elementary  and  Applied  Psychology.  —  EDWARD 

September  L.  THORNDIKE. 

No.  5  The  Economics  of  Manual  Training  {dealing  with  the  cost  of  the  equip- 

November  ment  and  the  maintenance  of  Manual  Training,  Domestic  Art  and 

Domestic  Science  in  each  of  the  grades  and  the  high  school). — 

Louis  ROUILLION. 

CONTENTS  OF  VOLUME  III— 1902 

No.  i  Horace  Mann  School:    Dedication  Number. 

January  Papers  by  PRESIDENT  DANIEL  C.  OILMAN,  SUPERINTENDENT  S.  T. 

DUTTON,  and  others,  on  Present-Day  Problems  in  Education. 

No.  a  Chemistry  and  Physics  in  the  Horace  Mann  High  School.  —  PROFESSOR 

March  JOHN  F.  WOODHULL. 

Nos.  3  and  4       Helps  for  the  Teaching  of  Casar.  —  PROFESSOR  GONZALEZ  LODGE,  and 
May,  September        MESSRS.  H.  H.  HUBBELL  and  WILLIAM  F.  LITTLE. 

No.  5  The  Speyer  School.    Part  I :  Its  history  and  purpose.  —  DEAN  RUSSELL, 

November  PROFESSOR  F.  M.  McMuRRY  and  MR.  JESSE  D.  BURKS. 

Back  numbers  for  sale  singly  or  by  the  hundred  at  regular  rates. 


80 


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